Mathematical Researches
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نتایجی در مورد صفر شدن برخی Ext-مدولهای خاص
Some results on the vanishing of certain Ext-modules
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alg
مقاله مستقل
Original Manuscript
فرض کنید R یک حلقه ی جابه جایی یک دار نوتری و a عضوی از R باشد.<br>
در این مقاله، به بررسی و مطالعه ی شرایطی روی حلقه ی R و ایده آل اصلی Ra می پردازیم
<div>که تحت آن داشته باشیم: 0≠(R/Ra,R)Ext<sup>1</sup><sub>R</sub> .</div>
<span style="font-size:11pt"><span style="line-height:16.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:12.0pt"><span style="font-family:"Times New Roman",serif">Introduction</span></span></b></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:normal"><span style="tab-stops:right 13.5pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> Let </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be a commutative Noetherian ring with identity, and let </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be an element of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Assume that either (a) </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><i> </i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">is an integral domain, or (b) </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><i> </i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">is a </span></span><span style="font-size:10.0pt"><span style="font-family:NimbusRomNo9L-Regu">Cohen–Macaulay</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> ring. <span style="color:#151515">It is well known that if</span> </span></span><m:omath><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>ht</m:r><m:r><i> </i></m:r><m:r><i>Ra</i></m:r><m:r><i>=1</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, then </span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></i></m:den></m:f><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>R</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>≠0.</m:r></span></span></i></m:omath> <span style="font-family:"Times New Roman",serif">So, it is natural to ask the following questions:</span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Question 1:</span></span></b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> Let </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be a commutative Noetherian ring with identity, and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be an element of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> such that </span></span><m:omath><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>ht</m:r><m:r><i> </i></m:r><m:r><i>Ra</i></m:r><m:r><i>=1</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Then, is it true that, </span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></i></m:den></m:f><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>R</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>≠0</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">?<span style="color:#151515"></span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="color:#151515">or </span></span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">in more general situations</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="color:#151515">,</span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Question 2:</span></span></b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> Let </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be a commutative Noetherian ring with identity, and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be an element of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Under what conditions is </span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></i></m:den></m:f><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>R</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>≠0</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">?<span style="color:#151515"></span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">The aim of this paper is to answer these questions.</span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:12.0pt"><span style="font-family:"Times New Roman",serif">Results and discussion</span></span></b></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"> In this paper, first we show that </span></span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:den></m:f></m:e></m:d><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>≅</m:r></span></span></span></i><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>∘</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>:</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>∘</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>:</m:r></span></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:sub></m:ssub><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></span></i></m:e></m:d></m:e></m:d></m:num><m:den><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></span></i></m:den></m:f></m:omath><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"> and next we give some conditions which </span></span></span><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif">guarantee that </span></span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></span></i></m:den></m:f><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"><m:r>≠</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>0</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif">. In addition we give some examples to illustrate these results. </span></span></span><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif">We denote the set of minimal members of associated primes of </span></span></span><m:omath><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"> by </span></span></span><m:omath><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>mAss</m:r><m:r><i>(</i></m:r><m:r><i>R</i></m:r><m:r><i>).</i></m:r></span></span></span></m:omath><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"> Let </span></span></span><m:omath><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>0 =</m:r></span></span></span></i><m:nary><m:narypr><m:chr m:val="⋂"><m:limloc m:val="undOvr"><m:suphide m:val="on"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:suphide></m:limloc></m:chr></m:narypr><m:sub><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>p</m:r><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>∈</m:r></span></span></span></i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ass</m:r><m:r><i> </i></m:r></span></span></span><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></span></i></m:sub><m:sup></m:sup><m:e><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>q</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>(</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>p</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></span></i></m:e></m:nary><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr> </m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"> be a reduced primary decomposition of the zero ideal. Set </span></span></span><m:omath><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>N</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r></span></span></span></i><m:nary><m:narypr><m:chr m:val="⋂"><m:limloc m:val="undOvr"><m:suphide m:val="on"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:suphide></m:limloc></m:chr></m:narypr><m:sub><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>p</m:r><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>∈</m:r></span></span></span></i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>mAss</m:r><m:r><i>(</i></m:r></span></span></span><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r><m:r>)</m:r></span></span></span></i></m:sub><m:sup></m:sup><m:e><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>q</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>(</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>p</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>)</m:r></span></span></span></i></m:e></m:nary></m:omath><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif">. </span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="tab-stops:right 9.0pt 13.5pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"Times New Roman",serif"> The following conclusions were drawn from this research.</span></span></span></span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">1. Let </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be a Noetherian ring and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be an element of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> such that </span></span><m:omath><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>ht</m:r><m:r><i> </i></m:r><m:r><i>Ra</i></m:r><m:r><i>=1</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">.</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> Then, </span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></i></m:den></m:f><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>R</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>≠0</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, if either of the following conditions holds:</span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> a) </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>aN</m:r><m:r>=0</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">;</span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> b) </span></span><m:omath><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>2</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>N</m:r><m:r>=0 </m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">and </span></span><m:omath><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>0</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>:</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>⊆</m:r><m:r>Ra</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">.</span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">2. Let </span></span><m:omath><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r><m:r>,</m:r><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>m</m:r></span></span></i></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be a Noetherian local ring and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>a</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be an element of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> such that </span></span><m:omath><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>ht</m:r><m:r><i> </i></m:r><m:r><i>Ra</i></m:r><m:r><i>=1</i></m:r></span></span></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">.</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> Then </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">we have </span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></i></m:den></m:f><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r><m:r>R</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>≠0</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, if either of the following holds:</span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> a) </span></span><m:omath><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>N</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>2</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=0</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">;</span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> b) </span></span><m:omath><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>0</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>:</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>N</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>⊈</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>m</m:r></span></span></i></m:e><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>2</m:r></span></span></i></m:sup></m:ssup></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">.</span></span></span></span><br>
<span style="line-height:16.0pt"><span style="text-autospace:none"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">3. Let </span></span><m:omath><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="fraktur"><m:sty m:val="i"></m:sty></m:scr></m:rpr>m</m:r></span></span></i></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be a Noetherian local ring such that </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>v</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><m:r>≤</m:r></span></span></i><m:func><m:funcpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:funcpr><m:fname><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>dim</m:r></span></span></m:fname><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:e></m:func><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>+1</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Then, </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">we have </span></span><m:omath><m:ssubsup><m:ssubsuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubsuppr><m:e><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Ext</m:r></span></span></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:sub><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>1</m:r></span></span></i></m:sup></m:ssubsup><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:f><m:fpr><m:type m:val="lin"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:type></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:num><m:den><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></i></m:den></m:f><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>,</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>R</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><m:r>≠</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>0</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> for any principal ideal </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Ra</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> of height one.</span></span><b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></b></span></span></span></span><br>
<br>
حلقه های نوتری, Ext-مدول ها, ایده آل های اصلی.
Noetherian rings, Ext-modules, principal ideals
675
681
http://mmr.khu.ac.ir/browse.php?a_code=A-10-938-1&slc_lang=fa&sid=1
Alireza
Nazari
علیرضا
نظری
nazari.ar@lu.ac.ir
10031947532846005769
10031947532846005769
Yes
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