• 대한수학회논문집
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• 제25권3호
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• pp.343-347
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• 2010

In this paper we study the connection between 2-primal rings, (S,1)-rings and related conditions. And we investigate some condition which is the special case of pseudo symmetric. We also study the condition (KJ) which is given by J. Y. Kim and H. L. Jin. We introduce some condition and we prove that our condition is equivalent to the condition (KJ) when it is an (S,1)-ring.

• 호남수학학술지
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• 제41권3호
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• pp.449-461
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• 2019

The aim of this paper is to extend the concept of quasi and bi-ideals from left almost semigroups to left almost rings which are the generalization of one sided ideals. Further, we discuss quasi and bi-ideals in regular left almost rings and intra regular left almost rings. We then explore many interesting and elegant properties of quasi and bi-ideals.

• Kyungpook Mathematical Journal
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• 제61권3호
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• pp.461-472
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• 2021

We study a new ring property called axis-commutativity. Axis-commutative rings are seated between commutative rings and duo rings and are a generalization of division rings. We investigate the basic structure and several extensions of axis-commutative rings.

• 대한수학회보
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• 제44권3호
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• pp.407-414
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• 2007

In [12], McCoy proved that if R is a commutative ring, then whenever g(x) is a zero-divisor in R[x], there exists a nonzero c $\in$ R such that cg(x) = 0. In this paper, first we extend this result to monoid rings. Then for a monoid M, we give some examples of M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings. Every reduced ring is M-quasi-Armendariz for any unique product monoid M and any strictly totally ordered monoid $(M,\;{\leq})$. Also $T_4(R)$ is M-quasi-Armendariz when R is reduced and M-Armendariz.

• 대한수학회지
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• 제42권2호
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• pp.353-363
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• 2005

For a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$, we introduce ($\alpha$, $\delta$)-skew Armendariz rings which are a generalization of $\alpha$-rigid rings and Armendariz rings, and investigate their properties. A semi prime left Goldie ring is $\alpha$-weak Armendariz if and only if it is $\alpha$-rigid. Moreover, we study on the relationship between the Baerness and p.p. property of a ring R and these of the skew polynomial ring R[x; $\alpha$, $\delta$] in case R is ($\alpha$, $\delta$)-skew Armendariz. As a consequence we obtain a generalization of [11], [14] and [16].

• Kyungpook Mathematical Journal
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• 제54권1호
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• pp.65-72
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• 2014

A ring R is called weakly semicommutative ring if for any a, $b{\in}R^*$ = R\{0} with ab = 0, there exists $n{\geq}1$ such that either an $a^n{\neq}0$ and $a^nRb=0$ or $b^n{\neq}0$ and $aRb^n=0$. In this paper, many properties of weakly semicommutative rings are introduced, some known results are extended. Especially, we show that a ring R is a strongly regular ring if and only if R is a left SF-ring and weakly semicommutative ring.

• 대한수학회보
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• 제46권1호
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• pp.135-146
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• 2009

In the present note we study the properties of weak Armendariz rings, and the connections among weak Armendariz rings, Armendariz rings, reduced rings and IFP rings. We prove that a right Ore ring R is weak Armendariz if and only if so is Q, where Q is the classical right quotient ring of R. With the help of this result we can show that a semiprime right Goldie ring R is weak Armendariz if and only if R is Armendariz if and only if R is reduced if and only if R is IFP if and only if Q is a finite direct product of division rings, obtaining a simpler proof of Lee and Wong's result. In the process we construct a semiprime ring extension that is infinite dimensional, from given any semi prime ring. We next find more examples of weak Armendariz rings.

• 대한수학회지
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• 제56권3호
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• pp.723-738
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• 2019

A property of reduced rings is proved in relation with centers, and our argument in this article is spread out based on this. It is also proved that the Wedderburn radical coincides with the set of all nilpotents in symmetric-over-center rings, implying that the Jacobson radical, all nilradicals, and the set of all nilpotents are equal in polynomial rings over symmetric-over-center rings. It is shown that reduced rings are reversible-over-center, and that given reversible-over-center rings, various sorts of reversible-over-center rings can be constructed. The structure of radicals in reversible-over-center and symmetric-over-center rings is also investigated.

• 대한수학회논문집
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• 제37권2호
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• pp.327-335
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• 2022

Let R be a commutative ring with identity. We call the ring R to be an almost weakly finite conductor if for any two elements a and b in R, there exists a positive integer n such that aⁿR ∩ bⁿR is finitely generated. In this article, we give some conditions for the trivial ring extensions and the amalgamated algebras to be almost weakly finite conductor rings. We investigate the transfer of these properties to trivial ring extensions and amalgamation of rings. Our results generate examples which enrich the current literature with new families of examples of nonfinite conductor weakly finite conductor rings.

• 대한수학회논문집
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• 제39권2호
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• pp.373-397
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• 2024

Lambek introduced the concept of symmetric rings to expand the commutative ideal theory to noncommutative rings. In this study, we propose an extension of symmetric rings called strongly α-symmetric rings, which serves as both a generalization of strongly symmetric rings and an extension of symmetric rings. We define a ring R as strongly α-symmetric if the skew polynomial ring R[x; α] is symmetric. Consequently, we provide proofs for previously established outcomes regarding symmetric and strongly symmetric rings, directly derived from the results we have obtained. Furthermore, we explore various properties and extensions of strongly α-symmetric rings.