Volume 7, Issue 1 (Vol.7, No. 1, 2021)
mmr 2021, 7(1): 127-132 |
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Taherifar A. On lattice of Basic Z-Ideals. mmr 2021; 7 (1) :127-132
URL: http://mmr.khu.ac.ir/article-1-2961-en.html
URL: http://mmr.khu.ac.ir/article-1-2961-en.html
Yasouj University , ataherifar@yu.ac.ir
Abstract: (2202 Views)
For an f-ring 
with bounded inversion property, we show that 
, the set of all basic z-ideals of , partially ordered by inclusion is a bounded distributive lattice. Also, whenever is a semiprimitive ring, 
, the set of all basic 
-ideals of , partially ordered by inclusion is a bounded distributive lattice. Next, for an f-ring with bounded inversion property, we prove that is a complemented lattice and is a semiprimitive ring if and only if is a complemented lattice and is a reduced ring if and only if the base elements for closed sets in the space 
are open and is semiprimitive if and only if the base elements for closed sets in the space 
are open and is reduced. As a result, whenever 
(i.e., the ring of continuous functions), we have 
is a complemented lattice if and only if 
is a complemented lattice if and only if 
is a 
-space../files/site1/files/71/12.pdf
with bounded inversion property, we show that , the set of all basic z-ideals of , partially ordered by inclusion is a bounded distributive lattice. Also, whenever is a semiprimitive ring, , the set of all basic -ideals of , partially ordered by inclusion is a bounded distributive lattice. Next, for an f-ring with bounded inversion property, we prove that is a complemented lattice and is a semiprimitive ring if and only if is a complemented lattice and is a reduced ring if and only if the base elements for closed sets in the space are open and is semiprimitive if and only if the base elements for closed sets in the space are open and is reduced. As a result, whenever (i.e., the ring of continuous functions), we have is a complemented lattice if and only if is a complemented lattice if and only if is a -space../files/site1/files/71/12.pdf
Type of Study: Original Manuscript |
Subject:
alg
Received: 2019/06/8 | Revised: 2021/05/24 | Accepted: 2019/10/15 | Published: 2021/05/31 | ePublished: 2021/05/31
Received: 2019/06/8 | Revised: 2021/05/24 | Accepted: 2019/10/15 | Published: 2021/05/31 | ePublished: 2021/05/31
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