fa ایده‌آل‌های اثر و مدول‌های کوهن-مکالی ماکسیمال روی یک حلقه گرنشتاین جبر alg مقاله مستقل Original Manuscript <div style="text-align: justify;"><span dir="RTL" lang="FA" style="font-size:11.0pt"><span style="line-height:107%"><span b="" nazanin="" style="font-family:">در این مقاله به بررسی ایده‌آل‌های اثر از ضرب تانسوری دو مدول می‌پردازیم. همچنین برخی از نتایج شناخته شده را به کمک ایده‌آل‌های اثر تعمیم و برای برخی دیگر اثبات جدید ارائه می‌کنیم. علاوه بر این، بررسی ایده‌آل‌های اثر از مدول‌های کوهن-مکالی ماکسیمال روی حلقه‌های گرنشتاین جزو اهداف این مقاله است.</span></span></span></div> &nbsp; <div style="text-align: justify;"><span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span calibri="" style="font-family:"><span new="" roman="" style="font-family:" times="">&nbsp;&nbsp;&nbsp;&nbsp; All rings throughout this paper are commutative and Noetherian</span><span new="" roman="" style="font-family:" times="">. Semidualizing modules were studied independently by Foxby [4], Golod [5], and Vasconcelos [11]. A finite R-module C is called semidualizing if the natural homothety map </span><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:3.0pt"><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:99pt; height:14.25pt" > </span></span></span></span></span><span new="" roman="" style="font-family:" times="">is an isomorphism and </span><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:3.0pt"><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:70.5pt; height:15pt" > </span></span></span></span></span><span new="" roman="" style="font-family:" times="">for all </span><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:3.0pt"><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAATCAIAAAB+9pigAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAARhJREFUSEvtVDsWwiAQJJ6FWPg8AZ7AZ2Nla4dlLmDpAbC0TGtlI55AT5AqcBfcJWBMQgy+FztpEj67M+wMmxhjyC/H5JfJMfcfYLDCI5dIH3eLxI7FUVfg4CIckhPCpZu8PkpwLqRqL/fNMQkTeFwJ5vJ5gL4YJQWHsywGp0HSgwGARQvxfwONwan5Y6C/A2hAszvssVn6QTC6zE53Y/I1uWyhvrubdhVuxcyn1K3Q6dz+WZF1WbDNym8N4qg9OaRbL+KAjxBAX8+PGvpTgL6hS9ID2as8CxEqSn8zIP1yERSoa6Cm6jEa1M55tyWI7MSXnAcMGZPZc7EIAZtWLmKhS3z5DpCsdaT1dfV8kn+7Hup2Ize7LtwT4pIDPytJMzgAAAAASUVORK5CYII=" style="width:24pt; height:14.25pt" ></span></span></span></span></span><span new="" roman="" style="font-family:" times="">. If a semidualizing R-module has finite injective dimension, it is called dualizing and is denoted by D. The ring itself is an example of a semidualizing R-module. Many researchers, in particular Sather-Wagstaff [12], have studied the semidualizing modules.</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span calibri="" style="font-family:"><span new="" roman="" style="font-family:" times="">&nbsp;&nbsp;&nbsp;&nbsp; Let M be an R-module. The trace ideal of M, denoted by </span><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:3.0pt"><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:33pt; height:14.25pt" ></span></span></span></span></span><span new="" roman="" style="font-family:" times="">, </span><span new="" roman="" style="font-family:" times="">is the sum of images of all homomorphisms from M to R. </span><span new="" roman="" style="font-family:" times="">Trace ideals have attracted the attention of many researchers in recent years. In particular, Herzog et al. [6] and Dao et al. [3] studied the trace ideals of canonical modules. Also, the trace ideals of semidualizing modules were studied in [1]. &nbsp;</span></span></span></span></span><br> <span style="font-size:11pt"><span style="line-height:normal"><span style="text-autospace:none"><span calibri="" style="font-family:"><span new="" roman="" style="font-family:" times="">&nbsp;&nbsp;&nbsp;&nbsp; In this paper, we study the trace ideals of tensor product of two arbitrary modules. We prove some known facts with a different approach via trace ideals. For example, let C and </span><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:3.0pt"><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAATCAIAAAAvYqvDAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAK1JREFUOE9j/P//PwNxgIk4ZSBVA6Z0ezojo/XEOygOuDMx3ZoRBqwnbod4486tKwxWYd4qCKVAvaqrGGpuA0Pk/+0JVkA1qhClt68dS6vJVwGyQFIQOasJYHVQfto2CHNbGgOUCVYK5CIrhGmAaEqDaQIpxaMSWRcwXOHuJhAbQKVAdzPoqIHcjR+QEFtAt6J4//a2CVZWMI8guxSYqqAhAgpICMCl8j/jEEqEAH49y3N+73iUAAAAAElFTkSuQmCC" style="width:10.5pt; height:14.25pt" > </span></span></span></span></span><span new="" roman="" style="font-family:" times="">be two semidualizing</span><span new="" roman="" style="font-family:" times=""> R-modules. We show that </span><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:3.0pt"><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:39pt; height:14.25pt" > </span></span></span></span></span><span new="" roman="" style="font-family:" times="">is projective if and only if C and&nbsp;</span><span style="font-size:11.0pt"><span style="line-height:107%"><span calibri="" style="font-family:"><span style="position:relative"><span style="top:3.0pt"><img alt="" id="_x0000_i1025" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAATCAIAAAAvYqvDAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAK1JREFUOE9j/P//PwNxgIk4ZSBVA6Z0ezojo/XEOygOuDMx3ZoRBqwnbod4486tKwxWYd4qCKVAvaqrGGpuA0Pk/+0JVkA1qhClt68dS6vJVwGyQFIQOasJYHVQfto2CHNbGgOUCVYK5CIrhGmAaEqDaQIpxaMSWRcwXOHuJhAbQKVAdzPoqIHcjR+QEFtAt6J4//a2CVZWMI8guxSYqqAhAgpICMCl8j/jEEqEAH49y3N+73iUAAAAAElFTkSuQmCC" style="width:10.5pt; height:14.25pt" ></span></span></span></span></span><span new="" roman="" style="font-family:" times=""> are projective </span><span new="" roman="" style="font-family:" times="">R-modules of rank 1.</span><span new="" roman="" style="font-family:" times=""> Also, we study</span><span new="" roman="" style="font-family:" times=""> the trace ideals of maximal Cohen-Macaulay modules over a Gorenstein local ring.</span></span></span></span></span></div> <br> &nbsp; حلقه جابه جایی نوتری, ایده‌آل اثر, مدول نیم‌دوگانی, حلقه گرنشتاین, مدول کوهن-مکالی ماکسیمال, کلاس باس, مدول متعارف, مدول پروژکتیو, مدول آزاد. commutative Noetherian ring, semidualizing module, trace ideal, Gorenstein ring, maximal Cohen-Macaulay module, projective module, free module, tensor product, canonical module, Bass class. 111 121 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1664-1&slc_lang=fa&sid=1 Mohammad Bagherpoor محمد باقرپور bagherpour.mohammad.2019@gmail.com 10031947532846006407 10031947532846006407 Yes Kharazmi University, Tehran. Iran دانشگاه خوارزمی Abdoljavad Taherizadeh عبدالجواد طاهری زاده taheri@khu.ac.ir 10031947532846006408 10031947532846006408 No Kharazmi University, Tehran. Iran دانشگاه خوارزمی