• The Pure and Applied Mathematics
• /
• v.8 no.2
• /
• pp.145-152
• /
• 2001

Posner [Proc. Amer. Math. Soc. 8 (1957), 1093-1100] defined a derivation on prime rings and Herstein [Canad, Math. Bull. 21 (1978), 369-370] derived commutative property of prime ring with derivations. Recently, Bergen [Canad. Math. Bull. 26 (1983), 267-227], Bell and Daif [Acta. Math. Hunger. 66 (1995), 337-343] studied derivations in primes and semiprime rings. Also, in near-ring theory, Bell and Mason [Near-Rungs and Near-Fields (pp. 31-35), Proceedings of the conference held at the University of Tubingen, 1985. Noth-Holland, Amsterdam, 1987; Math. J. Okayama Univ. 34 (1992), 135-144] and Cho [Pusan Kyongnam Math. J. 12 (1996), no. 1, 63-69] researched derivations in prime and semiprime near-rings. In this paper, Posner, Bell and Mason＇s results are extended in prime near-rings with some conditions.

• Communications of the Korean Mathematical Society
• /
• v.38 no.3
• /
• pp.663-677
• /
• 2023

The purpose of this paper is to introduce a new class of rings containing the class of SFT-rings and contained in the class of rings with Noetherian prime spectrum. Let A be a commutative ring with unit and I be an ideal of A. We say that I is SFT if there exist an integer k ≥ 1 and a finitely generated ideal F ⊆ I of A such that x^k ∈ F for every x ∈ I. The ring A is said to be nonnil-SFT, if each nonnil-ideal (i.e., not contained in the nilradical of A) is SFT. We investigate the nonnil-SFT variant of some well known theorems on SFT-rings. Also we study the transfer of this property to Nagata's idealization and the amalgamation algebra along an ideal. Many examples are given. In fact, using the amalgamation construction, we give an infinite family of nonnil-SFT rings which are not SFT.

• Communications of the Korean Mathematical Society
• /
• v.36 no.1
• /
• pp.1-10
• /
• 2021

Let R and S be two commutative rings, J be an ideal of S and f : R → S be a ring homomorphism. The amalgamation of R and S along J with respect to f, denoted by R ⋈f J, is the special subring of R × S defined by R ⋈f J = {(a, f(a) + j) | a ∈ R, j ∈ J}. In this paper, we study some basic properties of a special kind of R ⋈f J-modules, called the amalgamation of M and N along J with respect to ��, and defined by M ⋈�� JN := {(m, ��(m) + n) | m ∈ M and n ∈ JN}, where �� : M → N is an R-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.

• Journal of the Korean Mathematical Society
• /
• v.44 no.4
• /
• pp.987-995
• /
• 2007

We investigate some results which concern the types of Noetherian local rings. In particular, we show that if r(Ap) ${\le}$ depth Ap + 1 for each prime ideal p of a quasi-unmixed Noetherian local ring A, then A is Cohen-Macaulay. It is also shown that the Kawasaki conjecture holds when dim A ${\le}$ depth A + 1. At the end, we deal with some analogous results for modules, which are derived from the results studied on rings.

• Kyungpook Mathematical Journal
• /
• v.51 no.1
• /
• pp.1-10
• /
• 2011

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

• Kyungpook Mathematical Journal
• /
• v.43 no.4
• /
• pp.499-502
• /
• 2003

The aim of the present paper is to establish for commutativity of rings with unity 1 satisfying one of the properties $(xy)^{k+1}=x^ky^{k+1}x$ and $(xy)^{k+1}=yx^{k+1}y^k$, for all x, y in R, and the mapping $x{\rightarrow}x^k$ is an anti-homomorphism where $k{\geq}1$ is a fixed positive integer.

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.4
• /
• pp.723-735
• /
• 2006

In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over $\mathbb{Z}_4$ is free if C is generated by a polynomial which divides $X^m-1$. In this paper, we prove that a necessary and sufficient condition for a cyclic code over $\mathbb{Z}_pk$ of length m to be free is that it is generated by a polynomial which divides $X^m-1$. We also show that this can be extended to finite local rings with a principal maximal ideal.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.1
• /
• pp.179-186
• /
• 2019

This article mainly concentrates on the open question whether a right self-injective ring R is necessary QF if $R/S_l$ is left Goldie. It is answered affirmatively under the condition $S_l{\subseteq}S_r$, where $S_l$ and $S_r$ denote the left socle and right socle of R respectively. And the original condition "right self-injective" can be weakened to "right CS and right P-injective". It is also proved that a semiperfect, left and right mininjective ring R is QF if $S_r{\subseteq}^{ess}$ $R_R$ and $R/S_l$ is left Goldie.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.4
• /
• pp.645-652
• /
• 2002

In this note we investigate some results which concern the types of local rings. In particular it is shown that if the type of a quasi-unmixed local ring A is less than or equal to depth A ＋ 1, and $\hat{A}_p$ is Cohen-Macaulay for every prime $p\neq\hat{m}$, then A is Cohen-Macaulay. (This implies the previously known result: if A satisfies $(S_{n-1})}$, where n is the type of a .ins A, then A is Cohen-Macaulay.)

• Bulletin of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.265-270
• /
• 1995

Let $K_1, \ldots, K_n$ be fields of transcendence degrees $t_1, \ldots, t_n$ respectively over a common subfield F. O'Carroll and Qureshi [7] conjectured that the tensor product $R = K_1 \otimes K_2 \otimes \ldots \otimes K_n$ is an equidimensional Hilbert ring and proved the conjecture in special cases. Trung proved the conjecture [9] and O'Carroll, Bowman and Howie [3,5] generalized the Trung's result in two directions and obtained two theorems stated below.