• 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.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.277-290
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• 2011

We show that the ${\theta}$-prime radical of a ring R is the set of all strongly ${\theta}$-nilpotent elements in R, where ${\theta}$ is an automorphism of R. We observe some conditions under which the ${\theta}$-prime radical of coincides with the prime radical of R. Moreover we characterize elements in prime radicals of skew Laurent polynomial rings, studying (${\theta}$, ${\theta}^{-1}$)-(semi)primeness of ideals of R.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.667-679
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• 2007

We study the prime ideal structure of the direct sum of directed system of rings indexed by a commutative semigroup which is nilpotent extension of a group.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.511-515
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• 2012

We first show that the semiprimeness, primeness, and reducedness can go up to up-monoid rings. By these results we can compute the lower nilradicals of up-monoid rings, from which the well-known fact of Amitsur and McCoy for the polynomial rings can be extended to up-monoid rings.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.527-534
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• 2005

We consider countable rings with ascending chain condition on right annihilators. We determine the structure of a countable right p-injective Baer ring, a countable semi prime quasi-Baer ring and a countable quasi-Baer biregular ring.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.553-565
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• 2009

In this paper we prove that every generalized Jordan triple higher derivation on a 2-torsion free semiprime ring is a generalized higher derivation. This extend the main result of [9] to the case of a semiprime ring.

• Journal of applied mathematics & informatics
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• v.42 no.1
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• pp.199-206
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• 2024

The goal of this study is to bring out the following conclusion: Let 𝓡 be a non-commutative prime ring of characteristic not two and 𝓓 be a bi-semiderivation on 𝓡 with a function 𝖋 (surjective). If 𝓓 acts as an endomorphism or as an anti-endomorphism, then 𝓓 = 0 on 𝓡.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.857-866
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• 2009

Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and $\mu$, $\nu$ be a pair of generalized derivations on R. If < $\mu^2(x)+\nu(x),\;x^n$ > = 0 for all x $\in$ R, then $\mu$ and $\nu$ are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!-torsion free prime ring with the center $C_R$ and d, g be a pair of derivations on R. If < $d^2(x)+g(x)$, $x^n$ > $\in$ $C_R$ for all x $\in$ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1281-1293
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• 2023

The purpose of this paper is to study the relationship between the structure of a factor ring R/P and the behavior of some derivations of R. More precisely, we establish a connection between the commutativity of R/P and derivations of R satisfying specific identities involving the prime ideal P. Moreover, we provide an example to show that our results cannot be extended to semi-prime ideals.