• East Asian mathematical journal
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• 제34권3호
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• pp.305-316
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• 2018

Let R be a ring with identity. A right ideal ideal I of a ring R is called ref lexive (resp. completely ref lexive) if $aRb{\subseteq}I$ implies that $bRa{\subseteq}I$ (resp. if $ab{\subseteq}I$ implies that $ba{\subseteq}I$) for any $a,\;b{\in}R$. R is called ref lexive (resp. completely ref lexive) if the zero ideal of R is a reflexive ideal (resp. a completely reflexive ideal). Let K(R) (called the ref lexive radical of R) be the intersection of all reflexive ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an reflexive ideal of a ring are obtained; (2) reflexive (resp. completely reflexive) property is Morita invariant; (3) For any ring R, we have $K(M_n(R))=M_n(K(R))$ where $M_n(R)$ is the ring of all n by n matrices over R; (4) For a ring R, we have $K(R)[x]{\subseteq}K(R[x])$; in particular, if R is quasi-Armendaritz, then R is reflexive if and only if R[x] is reflexive.

• 대한수학회논문집
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• 제24권2호
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• pp.161-169
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• 2009

In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring N, denoted by $\widehat{{\Gamma}_I(N)}$. It is shown that if I is a completely reflexive ideal of N, then every two vertices in $\widehat{{\Gamma}_I(N)}$ are connected by a path of length at most 3, and if $\widehat{{\Gamma}_I(N)}$ contains a cycle, then the core K of $\widehat{{\Gamma}_I(N)}$ is a union of triangles and rectangles. We have shown that if $\widehat{{\Gamma}_I(N)}$ is a bipartite graph for a completely semiprime ideal I of N, then N has two prime ideals whose intersection is I.