• East Asian mathematical journal
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• v.36 no.3
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• pp.417-424
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• 2020

Huang et al. proved that the n by n upper triangular matrix ring over a domain is weakly reversible-over-center by using the property of regular matrices. In this article we provide a concrete proof which is able to be available in the related study of centers. Next we extend an example of weakly reversible-over-center, which was argued by Huang et al., to the general case.

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.93-108
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• 2011

A ring R is called left weakly np- injective if for each non-nilpotent element a of R, there exists a positive integer n such that any left R- homomorphism from $Ra^n$ to R is right multiplication by an element of R. In this paper various properties of these rings are first developed, many extending known results such as every left or right module over a left weakly np- injective ring is divisible; R is left seft-injective if and only if R is left weakly np-injective and $_RR$ is weakly injective; R is strongly regular if and only if R is abelian left pp and left weakly np- injective. We next introduce the concepts of left weakly pp rings and left weakly C2 rings. In terms of these rings, we give some characterizations of (von Neumann) regular rings such as R is regular if and only if R is n- regular, left weakly pp and left weakly C2. Finally, the relations among left C2 rings, left weakly C2 rings and left GC2 rings are given.

• East Asian mathematical journal
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• v.22 no.1
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• pp.61-69
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• 2006

A near-ring N is reduced if, for $a{\in}N,\;a^2=0$ implies a=0, and N is left strongly regular if for all $a{\in}N$ there exists $x{\in}N$ such that $a=xa^2$. Mason introduced this notion and characterized left strongly regular zero-symmetric unital near-rings. Several authors ([2], [5], [7]) studied these properties in near-rings. Reddy and Murty extended some results in Mason to the non-zero symmetric case. In this paper, we will define a concept of strong reducedness and investigate a relation between strongly reduced near-rings and left strongly regular near-rings.

• East Asian mathematical journal
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• v.36 no.3
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• pp.337-347
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• 2020

Let R be a ring with unity, and let I be an ideal of R. Then R is called I-semiregular if for every a ∈ R there exists b ∈ R such that ab is an idempotent of R and a - aba ∈ I. In this paper, basic properties of I-semiregularity are investigated, and some equivalent conditions to the primitivity of e are observed for an idempotent e of an I-semiregular ring R such that I∩eR = (0). For an abelian regular ring R with the ascending chain condition on annihilators of idempotents of R, it is shown that R is isomorphic to a direct product of a finite number of division rings, as a consequence of the observations.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1177-1190
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• 2022

Let R be a commutative ring with identity. We call the ring R to be an almost quasi-coherent ring if for any finite set of elements α₁, …, α_p and a of R, there exists a positive integer m such that the ideals $\bigcap{_{i=1}^{p}}\;R{\alpha}^m_i$ and Ann_R(α^m) are finitely generated, and to be almost von Neumann regular rings if for any two elements a and b in R, there exists a positive integer n such that the ideal (αⁿ, bⁿ) is generated by an idempotent element. This paper establishes necessary and sufficient conditions for the Nagata's idealization and the amalgamated algebra to inherit these notions. Our results allow us to construct original examples of rings satisfying the above-mentioned properties.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1629-1643
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• 2016

Let R be a ring with identity, X be the set of all nonzero, nonunits of R and G be the group of all units of R. A ring R is called unit-duo ring if $[x]_{\ell}=[x]_r$ for all $x{\in}X$ where $[x]_{\ell}=\{ux{\mid}u{\in}G\}$ (resp. $[x]_r=\{xu{\mid}u{\in}G\}$) which are equivalence classes on X. It is shown that for a semisimple unit-duo ring R (for example, a strongly regular ring), there exist a finite number of equivalence classes on X if and only if R is artinian. By considering the zero divisor graph (denoted ${\tilde{\Gamma}}(R)$) determined by equivalence classes of zero divisors of a unit-duo ring R, it is shown that for a unit-duo ring R such that ${\tilde{\Gamma}}(R)$ is a finite graph, R is local if and only if diam(${\tilde{\Gamma}}(R)$) = 2.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.715-726
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• 2010

We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining $hs(Mat_n(R))\;=\;Mat_n(hs(R))$ for any ring R where hs(-) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunowicz's simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1887-1903
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• 2013

We continue the study of McCoy condition to analyze zero-dividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-${\pi}$-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-${\pi}$-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-${\pi}$-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-${\pi}$-McCoy rings by examining various sorts of ordinary ring extensions.

• 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.46 no.5
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• pp.873-884
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• 2009

We characterize QB-ideals of exchange rings by means of quasi-invertible elements and annihilators. Further, we prove that every $2\times2$ matrix over such ideals of a regular ring admits a diagonal reduction by quasi-inverse matrices. Prime exchange QB-rings are studied as well.