• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.37-43
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• 2008

The aim of the present paper is to introduce the concept "Finite dimension" in the theory of associative rings R with respect to two sided ideals. We obtain that if R has finite dimension on two sided ideals, then there exist uniform ideals $U_1,U_2,\ldots,U_n$ of R whose sum is direct and essential in R. The number n is independent of the choice of the uniform ideals $U_i$ and 'n' is called the dimension of R.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.653-661
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• 2010

Let R be a ring, and let $\Psi$(R) be the ideal generated by the set {x $\in$R | 1 + sxt $\in$ R is unit-regular for all s, t $\in$ R}. We show that $\Psi$(R) has "radical-like" property. It is proven that $\Psi$(R) has stable range one. Thus, diagonal reduction of matrices over such ideal is reduced.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.337-345
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• 2022

Let R ⊆ T be an extension of a commutative ring and S ⊆ R a multiplicative subset. We say that (R, T) is an S-accr (a commutative ring R is said to be S-accr if every ascending chain of residuals of the form (I : B) ⊆ (I : B²) ⊆ (I : B³) ⊆ ⋯ is S-stationary, where I is an ideal of R and B is a finitely generated ideal of R) pair if every ring A with R ⊆ A ⊆ T satisfies S-accr. Using this concept, we give an S-version of several different known results.

• Journal of the Korean Mathematical Society
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• v.42 no.5
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• pp.933-948
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• 2005

Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)$\bigcap$V(ann$\_{R}$(M))$\rightarrow$Spec$\_{R}$(M) and in particular, there exists a bijection: N(M)$\bigcap$Max(R)$\rightarrow$Max$\_{R}$(M), (2) N(M) $\bigcap$ V(ann$\_{R}$(M)) = Supp(M) $\bigcap$ V(ann$\_{R}$(M)), and (3) for every ideal I of R, The ideal $\theta$(M) = $\sum$$\_{m(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P $\in$ Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P$^{n}$M$\neq$P$^{n+1}$ for every positive integer n, then $\bigcap$$^{$\_{n=1}$(P$^{n}$ + ann$\_{R}$(M)) $\in$ V(ann$\_{R}$(M)) = Supp(M) $\subseteq$ N(M).

• 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 $n{\geq} 2$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1401-1407
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• 2021

Let R be a commutative G-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal I of R is said to be graded radically principal if Grad(I) = Grad(〈c〉) for some homogeneous c ∈ R, where Grad(I) is the graded radical of I. The graded ring R is said to be graded radically principal if every graded ideal of R is graded radically principal. We study graded radically principal rings. We prove an analogue of the Cohen theorem, in the graded case, precisely, a graded ring is graded radically principal if and only if every graded prime ideal is graded radically principal. Finally we study the graded radically principal property for the polynomial ring R[X].

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1387-1408
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• 2022

Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :_R M)∩S = ∅ to be weakly S-prime if there exists s ∈ S such that whenever a ∈ R and m ∈ M with 0 ≠ am ∈ N, then either sa ∈ (N :_R M) or sm ∈ N. Many properties, examples and characterizations of weakly S-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly S-prime.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.21-28
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• 2015

Let R be a prime ring with center Z, I a nonzero ideal of R, and ${\sigma}$ a non-trivial automorphism of R such that $\{(x{\circ}y)^{\sigma}-(x{\circ}y)\}^n{\in}Z$ for all $x,y{\in}I$. If either char(R) > n or char (R) = 0, then R satisfies $s_4$, the standard identity in 4 variables.

• Journal of the Korean Mathematical Society
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• v.42 no.1
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• pp.53-63
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• 2005

Let R be a ring R and ${\sigma}$ be an endomorphism of R. R is called ${\sigma}$-rigid (resp. reduced) if $a{\sigma}r(a) = 0 (resp{\cdot}a^2 = 0)$ for any $a{\in}R$ implies a = 0. An ideal I of R is called a ${\sigma}$-ideal if ${\sigma}(I){\subseteq}I$. R is called ${\sigma}$-quasi-Baer (resp. right (or left) ${\sigma}$-p.q.-Baer) if the right annihilator of every ${\sigma}$-ideal (resp. right (or left) principal ${\sigma}$-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[$x;{\sigma}$] of a ring R is investigated as follows: For a ${\sigma}$-rigid ring R, (1) R is ${\sigma}$-quasi-Baer if and only if A is quasi-Baer if and only if A is $\={\sigma}$-quasi-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$ (2) R is right ${\sigma}$-p.q.-Baer if and only if R is ${\sigma}$-p.q.-Baer if and only if A is right p.q.-Baer if and only if A is p.q.-Baer if and only if A is $\={\sigma}$-p.q.-Baer if and only if A is right $\={\sigma}$-p.q.-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.295-305
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• 2005

In this paper, we investigate exchange ideals and get some new characterization of exchange rings. It is shown that an ideal I of a ring R is an exchange ideal if and only if so is $QM_2$(I). Also we observe that every exchange ideal can be characterized by exchange elements.