• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.13-21
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• 1997

In recent years M. Hochster and C. Huneke introduced the notions of tight closure of an ideal and of the weak F-regularity of a ring of positive prime characteristic. Here 'F' stands for Frobenius. This notion enabled us to play an important role in a commutative ring theory, and other related topics.

• Journal of the Chungcheong Mathematical Society
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• v.7 no.1
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• pp.1-11
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• 1994

The theory of inverse semigroups has many features in common with the theory of groups. Many different properties of semigroup become the same condition on ring. In this paper, we want to find the properties of semigroups which is preserved by the properties of ring. Also we find that many different properties become the equivalent conditions.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.439-469
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• 2006

Intrinsic product of intuitionistic fuzzy sets are considered. Using this, characterizations of intuitionistic fuzzy subrings/ideals are discussed. The notions of intuitionistic fuzzy quasi ideals and intuitionistic fuzzy bi-ideals are introduced. Characterizations of regular rings are provided.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.569-575
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• 2007

Let M be a nonzero module. M has the property that every submodule of M lies over a direct summand of M. We study some properties of such a module. The endomorphism ring of such a module is also studied. The relationships of such a module to the semi-regular modules, and to the semi-perfect modules are described.

• East Asian mathematical journal
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• v.26 no.3
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• pp.397-401
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• 2010

We investigate some properties of strongly reduced near-rings and left regular near-rings, also show that a strongly reduced near-ring is reduced. Next, we will characterize that reducibility, strong reducibility and left regularity in near-rings.

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.755-762
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• 2006

For a commutative ring R and an injective cogenerator E in the category of R-modules, we characterize von Neumann regular rings and semisimple artinian rings in terms of the properties of dual modules with respect to E.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.813-822
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• 2014

A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.

• Kyungpook Mathematical Journal
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• v.62 no.2
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• pp.213-227
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• 2022

We discuss the relation between units and nilpotents of a ring, concentrating on the transitivity of units on nilpotents under regular group actions. We first prove that for a ring R, if U(R) is right transitive on N(R), then Köthe's conjecture holds for R, where U(R) and N(R) are the group of all units and the set of all nilpotents in R, respectively. A ring is called right UN-transitive if it satisfies this transitivity, as a generalization, a ring is called unilpotent-IFP if aU(R) ⊆ N(R) for all a ∈ N(R). We study the structures of right UN-transitive and unilpotent-IFP rings in relation to radicals, NI rings, unit-IFP rings, matrix rings and polynomial rings.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.83-91
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• 2023

A ring R is called a UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We obtain results about lifting of conjugate idempotents and unit regular elements modulo an ideal I of a UN ring R. Matrix rings over UN rings are discussed and it is obtained that for a commutative ring R, a matrix ring M_n(R) is UN if and only if R is UN. Lastly, UN group rings are investigated and we obtain the conditions on a group G and a field K for the group algebra KG to be UN. Then we extend the results obtained for KG to the group ring RG over a ring R (which may not necessarily be a field).

• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.169-177
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• 2003

Recently, K. Watanabe Showed that the Hilbert-Kunz multiplicity of a toric ring is a rational number. In this paper we give an explicit formula to compute the Hilbert-Kunz multiplicity of two-dimensional toric rings. This formula also shows that the Hilbert-Kunz multiplicity of a two-dimensional non-regular toric ring is at least 3/2.