• Korean Journal of Mathematics
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• v.5 no.2
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• pp.135-139
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• 1997

We introduce the concepts of weakly prime and weakly semi-prime ideals in po-${\Gamma}$-semigroup and give some characterizations of weakly prime and weakly semi-prime ideals of po-${\Gamma}$-semigroups analogous to the characterizations of weakly prime and weakly semi-prime ideals of po-semigroups considered by N. Kehayopulu.

• East Asian mathematical journal
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• v.19 no.2
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• pp.233-240
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• 2003

Xie and Wu introduced an m-system in a po-semigroup. Kehayopulu gave characterizations of weakly prime ideals of po-semigroups and Lee and Kwon add two characterizations for weakly prime ideals. In this paper, we give a characterization of weakly prime ideals and a characterization of weakly semi-prime ideals in po-semigroups using m-system and n-system, respectively

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1183-1198
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• 2013

We will introduce and study the notion of prime bases for weakly prime submodules and utilize them to derive some formulas on the weak radical of submodules of a module. In particular, we will show that every one dimensional integral domain weakly satisfies the radical formula and state some necessary conditions on local integral domains which are semi-compatible or satisfy the radical formula and also on Noetherian rings which weakly satisfy the radical formula.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.277-285
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• 2017

In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

• 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.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1167-1182
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• 2016

Let M be an R-module, where R is a commutative ring with identity 1 and let G(V,E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$. We show that M over R is finite if and only if the metric dimension of the graph $ann_f({\Gamma}(M_R))$ is finite. We further show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if M is a prime-multiplication-like R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if $$M{\sim_=}R$$. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M) is a prime ideal and Soc(M) = 0.