• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.781-786
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• 2013

Let * be a star-operation on an integral domain R. We show that if R is a $*_{\omega}$-Noetherian domain, then quasi-$*_{\omega}$-invertible prime $*_{\omega}$-ideals of R are minimal, and prime ideals of R minimal over a $*_{\omega}$-invertible $*_{\omega}$-ideal are minimal.

• Korean Journal of Mathematics
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• v.21 no.2
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• pp.197-201
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• 2013

Let D be an integral domain with quotient field K, * a star-operation on D, $GV^*(D)$ the set of nonzero finitely generated ideals J of D such that $J_*=D$, and $*_{\omega}$ a star-operation on D defined by $I_{*_{\omega}}=\{x{\in}K{\mid}Jx{\subseteq}I\;for\;some\;J{\in}GV^*(D)\}$ for all nonzero fractional ideals I of D. In this article, we give a simple proof of Hilbert basis theorem for $*_{\omega}$-Noetherian domains.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.379-395
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• 2018

Assume that R is a commutative ring with non-zero identity which is not an integral domain. An ideal I of R is called an annihilating ideal if there exists a non-zero element $a{\in}R$ such that Ia = 0. S. Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of R, denoted by ${\Omega}(R)$, as the graph with the vertex-set $A(R)^*$, the set of all non-zero annihilating ideals of R, and two distinct vertices I and J are adjacent if I + J is an annihilating ideal. In this paper, we study the relations between the diameters of ${\Omega}(R)$ and ${\Omega}(R[x])$. Also, we study the relations between the diameters of ${\Omega}(R)$ and ${\Omega}(R[[x]])$, whenever R is a Noetherian ring. In addition, we investigate the relations between the diameters of this graph and the zero-divisor graph. Moreover, we study some combinatorial properties of ${\Omega}(R)$ such as domination number and independence number. Furthermore, we study the complement of this graph.