• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1123-1139
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• 2017

The aim in this note is to describe some classes of rings in relation to factorization by prime radical, upper nilradical, and Jacobson radical. We introduce the concepts of tpr ring, tunr ring, and tjr ring in the process, respectively. Their ring theoretical structures are investigated in relation to various sorts of factor rings and extensions. We also study the structure of noncommutative tpr (tunr, tjr) rings of minimal order, which can be a base of constructing examples of various ring structures. Various sorts of structures of known examples are studied in relation with the topics of this note.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1957-1972
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• 2013

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. In this note we study the structure of idempotents satisfying the reflexive property and introduce reflexive-idempotents-property (simply, RIP) as a generalization. It is proved that the RIP can go up to polynomial rings, power series rings, and Dorroh extensions. The structure of non-Abelian RIP rings of minimal order (with or without identity) is completely investigated.

• 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.

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.489-501
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• 2015

We study the structure of right duo ring property when it is restricted within the group of units, and introduce the concept of right unit-duo. This newly introduced property is first observed to be not left-right symmetric, and we examine several conditions to ensure the symmetry. Right unit-duo rings are next proved to be Abelian, by help of which the class of noncommutative right unit-duo rings of minimal order is completely determined up to isomorphism. We also investigate some properties of right unit-duo rings which are concerned with annihilating conditions.