• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.289-309
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• 2019

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.603-615
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• 2019

Let R be an associative unital ring with an endomorphism α and α-derivation δ. The constant products of elements in Ore extension rings, when the coefficient ring is reversible, is investigated. We show that if f(x) = ∑ⁿ_i=0 a_ixⁱ and g(x) = ∑^m_j=0 b_jx^j be nonzero elements in Ore extension ring R[x; α, δ] such that g(x)f(x) = c ∈ R, then there exist non-zero elements r, a ∈ R such that rf(x) = ac, when R is an (α, δ)-compatible ring which is reversible. Among applications, we give an exact characterization of the unit elements in R[x; α, δ], when the coeficient ring R is (α, δ)-compatible. Furthermore, it is shown that if R is a weakly 2-primal ring which is (α, δ)-compatible, then J(R[x; α, δ]) = N iℓ(R)[x; α, δ]. Some other applications and examples of rings with this property are given, with an emphasis on certain classes of NI rings. As a consequence we obtain generalizations of the many results in the literature. As the final part of the paper we construct examples of rings that explain the limitations of the results obtained and support our main results.

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.119-132
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• 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for $a,b{\in}R$. In this paper, we study an extension of a reversible ring with its endomorphism. An endomorphism ${\alpha}$ of a ring R is called strong right (resp., left) reversible if whenever $a{\alpha}(b)=0$ (resp., ${\alpha}(a)b=0$) for $a,b{\in}R$, ba = 0. A ring R is called strong right (resp., left) ${\alpha}$-reversible if there exists a strong right (resp., left) reversible endomorphism ${\alpha}$ of R, and the ring R is called strong ${\alpha}$-reversible if R is both strong left and right ${\alpha}$-reversible. We investigate characterizations of strong ${\alpha}$-reversible rings and their related properties including extensions. In particular, we show that every semiprime and strong ${\alpha}$-reversible ring is ${\alpha}$-rigid and that for an ${\alpha}$-skew Armendariz ring R, the ring R is reversible and strong ${\alpha}$-reversible if and only if the skew polynomial ring $R[x;{\alpha}]$ of R is reversible.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.247-261
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• 2015

We study the connections between idempotents and zero-divisors in several kinds of ring theoretic properties. We next study several ring theoretic properties and examples related to reversible rings.

• Communications of the Korean Mathematical Society
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• v.25 no.3
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• pp.349-364
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• 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, $b\;{\in}\;R$. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and $\alpha$ an endomorphism of R, we say that R is right (resp., left) $\alpha$-shifting if whenever $a{\alpha}(b)\;=\;0$ (resp., $\alpha{a)b\;=\;0$) for a, $b\;{\in}\;R$, $b{\alpha}{a)\;=\;0$ (resp., $\alpha(b)a\;=\;0$); and the ring R is called $\alpha$-shifting if it is both left and right $\alpha$-shifting. We investigate characterizations of $\alpha$-shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism $\alpha$ of a ring R, R is right (resp., left) $\alpha$-shifting if and only if Q(R) is right (resp., left) $\bar{\alpha}$-shifting, whenever there exists the classical right quotient ring Q(R) of R.

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.455-466
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• 2010

For an endomorphism $\alpha$ of a ring R, we introduce the weak $\alpha$-skew Armendariz rings which are a generalization of the $\alpha$-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak $\alpha$-skew Armendariz if and only if for any n, the $n\;{\times}\;n$ upper triangular matrix ring $T_n(R)$ is weak $\bar{\alpha}$-skew Armendariz, where $\bar{\alpha}\;:\;T_n(R)\;{\rightarrow}\;T_n(R)$ is an extension of $\alpha$ If R is reversible and $\alpha$ satisfies the condition that ab = 0 implies $a{\alpha}(b)=0$ for any a, b $\in$ R, then the ring R[x]/($x^n$) is weak $\bar{\alpha}$-skew Armendariz, where ($x^n$) is an ideal generated by $x^n$, n is a positive integer and $\bar{\alpha}\;:\;R[x]/(x^n)\;{\rightarrow}\;R[x]/(x^n)$ is an extension of $\alpha$. If $\alpha$ also satisfies the condition that ${\alpha}^t\;=\;1$ for some positive integer t, the ring R[x] (resp, R[x; $\alpha$) is weak $\bar{\alpha}$-skew (resp, weak) Armendariz, where $\bar{\alpha}\;:\;R[x]\;{\rightarrow}\;R[x]$ is an extension of $\alpha$.

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.273-284
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• 2012

A ring R is called reversibly Armendariz if $b_ja_i=0$ for all $i$, $j$ whenever $f(x)g(x)=0$ for two polynomials $f(x)=\sum_{i=0}^{m}a_ix^i,\;g(x)=\sum_{j=0}^{n}b_jx^j$ over R. It is proved that a ring R is reversibly Armendariz if and only if its polynomial ring is reversibly Armendariz if and only if its Laurent polynomial ring is reversibly Armendariz. Relations between reversibly Armendariz rings and related ring properties are examined in this note, observing the structures of many examples concerned. Various kinds of reversibly Armendariz rings are provided in the process. Especially it is shown to be possible to construct reversibly Armendariz rings from given any Armendariz rings.

• Korean Journal of Mathematics
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• v.22 no.2
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• pp.279-288
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• 2014

The studies of reversible and 2-primal rings have done important roles in noncommutative ring theory. We in this note introduce the concept of quasi-reversible-over-prime-radical (simply, QRPR) as a generalization of the 2-primal ring property. A ring is called QRPR if ab = 0 for $a,b{\in}R$ implies that ab is contained in the prime radical. In this note we study the structure of QRPR rings and examine the QRPR property of several kinds of ring extensions which have roles in noncommutative ring theory.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.751-771
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• 2014

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that ${\pi}$-regular rings are strongly ${\pi}$-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.

• East Asian mathematical journal
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• v.34 no.5
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• pp.615-621
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• 2018

Let R be a ring with the unity 1, and let e be an idempotent of R. In this paper, we discuss some symmetric property for the set $\{(a_1,a_2,{\cdots},a_n){\in}R^n:a_1a_2{\cdots}a_n=e\}$. We here investigate some properties of those rings with such a symmetric property for an arbitrary idempotent e; some of our results turn out to generalize some known results observed already when n = 2 and e = 0, 1 by several authors. We also focus especially on the case when n = 3 and e = 1. As consequences of our observation, we also give some equivalent conditions to the commutativity for some classes of rings, in terms of the symmetric property.