• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.299-307
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• 2006

The following results ale extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every finitely generated left R-module M, $_R(M/Z(M))$ is projective, where Z(M) is the left singular submodule of $_{R}M$; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming [13] in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essential left ideal, then R is a left and right self-injective regular, left and right V-ring of bounded index.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.597-601
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• 2006

We introduce in this paper the concept of idempotent reflexive right ideals and concern with rings containing an injective maximal right ideal. Some known results for reflexive rings and right HI-rings can be extended to idempotent reflexive rings. As applications, we are able to give a new characterization of regular right self-injective rings with nonzero socle and extend a known result for right weakly regular rings.

• Journal of the Korean Mathematical Society
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• v.49 no.1
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• pp.31-47
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• 2012

The so-called Ding-Chen ring is an n-FC ring which is both left and right coherent, and has both left and right self FP-injective dimensions at most n for some non-negative integer n. In this paper, we investigate the classes of the so-called Ding projective, Ding injective and Gorenstein at modules and show that some homological properties of modules over Gorenstein rings can be generalized to the modules over Ding-Chen rings. We first consider Gorenstein at and Ding injective dimensions of modules together with Ding injective precovers. We then discuss balance of functors Hom and tensor.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.179-186
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• 2019

This article mainly concentrates on the open question whether a right self-injective ring R is necessary QF if $R/S_l$ is left Goldie. It is answered affirmatively under the condition $S_l{\subseteq}S_r$, where $S_l$ and $S_r$ denote the left socle and right socle of R respectively. And the original condition "right self-injective" can be weakened to "right CS and right P-injective". It is also proved that a semiperfect, left and right mininjective ring R is QF if $S_r{\subseteq}^{ess}$ $R_R$ and $R/S_l$ is left Goldie.

• Kyungpook Mathematical Journal
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• v.26 no.2
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• pp.137-148
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• 1986

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.223-234
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• 2003

This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $_{A}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $_{A}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1711-1723
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• 2013

P. V$\acute{a}$mos called a ring R 2-good if every element is the sum of two units. The ring of all $n{\times}n$ matrices over an elementary divisor ring is 2-good. A (right) self-injective von Neumann regular ring is 2-good provided it has no 2-torsion. Some of the earlier results known to us about 2-good rings (although nobody so called at those times) were due to Ehrlich, Henriksen, Fisher, Snider, Rapharl and Badawi. We continue in this paper the study of 2-good rings by several authors. We give some examples of 2-good rings and their related properties. In particular, it is shown that if R is an exchange ring with Artinian primitive factors and 2 is a unit in R, then R is 2-good. We also investigate various kinds of extensions of 2-good rings, including the polynomial extension, Nagata extension and Dorroh extension.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1053-1066
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• 2010

In this paper, several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring $M^*$ of endomorphisms of M and hence the relation between the prime (maximal) submodules of M and the prime (maximal) ideals of $M^*$ can be found. In particular, two classes of ideals of $M^*$ are discussed in this paper: one is of the form $G_{M^*}\;(M,\;N)\;=\;\{f\;{\in}\;M^*\;|\;f(M)\;{\subseteq}\;N\}$ and the other is of the form $G_{M^*}\;(N,\;0)\;=\;\{f\;{\in}\;M^*\;|\;f(N)\;=\;0\}$ for a submodule N of M.