• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1367-1383
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• 2021

This paper studies Young diagrams of symmetric and pseudo-symmetric numerical semigroups and describes new operations on Young diagrams as well as numerical semigroups. These provide new decompositions of symmetric and pseudo-symmetric semigroups into a numerical semigroup and its dual. It is also given exactly for what kind of numerical semigroup S, the semigroup ring 𝕜⟦S⟧ has at least one Gorenstein subring and has at least one Kunz subring.

• 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.

• East Asian mathematical journal
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• v.4
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• pp.87-99
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• 1988

• East Asian mathematical journal
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• v.1
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• pp.21-24
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• 1985

• Kyungpook Mathematical Journal
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• v.30 no.1
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• pp.75-79
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• 1990

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.

• Korean Journal of Mathematics
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• v.29 no.3
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• pp.483-492
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• 2021

This article concerns the property that for any element a in a ring, if a²ⁿ = aⁿ for some n ≥ 2 then a² = a. The class of rings with this property is large, but there also exist many kinds of rings without that, for example, rings of characteristic ≠2 and finite fields of characteristic ≥ 3. Rings with such a property is called reduced-over-idempotent. The study of reduced-over-idempotent rings is based on the fact that the characteristic is 2 and every nonzero non-identity element generates an infinite multiplicative semigroup without identity. It is proved that the reduced-over-idempotent property pass to polynomial rings, and we provide power series rings with a partial affirmative argument. It is also proved that every finitely generated subring of a locally finite reduced-over-idempotent ring is isomorphic to a finite direct product of copies of the prime field {0, 1}. A method to construct reduced-over-idempotent fields is also provided.

• Korean Journal of Mathematics
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• v.19 no.3
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• pp.255-261
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• 2011

Let D be an integral domain, S be a torsion-free grading monoid such that the quotient group of S is of type (0, 0, 0, ${\ldots}$), and D[S] be the semigroup ring of S over D. We show that D[S] is an H-domain if and only if D is an H-domain and each maximal t-ideal of S is a $v$-ideal. We also show that if $\mathbb{R}$ is the eld of real numbers and if ${\Gamma}$ is the additive group of rational numbers, then $\mathbb{R}[{\Gamma}]$ is not an H-domain.

• 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.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.915-923
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• 2022

The purpose of the present paper is to obtain commutativity of prime Γ-near-ring N with generalized derivations F and G with associated derivations d and h respectively satisfying one of the following conditions:(i) G([x, y]α = ±f(y)α(xoy)βγg(y), (ii) F(x)βG(y) = G(y)βF(x), for all x, y ∈ N, β ∈ Γ (iii) F(u)βG(v) = G(v)βF(u), for all u ∈ U, v ∈ V, β ∈ Γ,(iv) if 0 ≠ F(a) ∈ Z(N) for some a ∈ V such that F(x)αG(y) = G(y)αF(x) for all x ∈ V and y ∈ U, α ∈ Γ.