• Kyungpook Mathematical Journal
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• 제47권3호
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• pp.363-372
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• 2007

An element $a$ in a ring R is called left morphic if $$R/Ra{\simeq_-}1(a)$$. A ring is called left morphic if every element is left morphic. In this paper, an element $a$ in a ring R is called left ${\pi}$-morphic (resp. left G-morphic) if there exists a positive number $n$ such that $a^n$ (resp. $a^n{\neq}0$) is left morphic. A ring R is called left ${\pi}$-morphic (resp. left G-morphic) if every element is left ${\pi}$-morphic (resp. left G-morphic). The Morita invariance of left ${\pi}$-morphic (resp. left G-morphic) rings is discussed. Several relevant properties are proved. In particular, it is shown that a left Noetherian ring R with $M_4(R)$ left G-morphic or $M_2(R)$ left morphic is QF. Some known results of left morphic rings are extended to left G-morphic rings and left ${\pi}$-morphic rings.

• 대한수학회보
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• 제39권4호
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• pp.645-652
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• 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.)

• 대한수학회보
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• 제51권1호
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• pp.253-266
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• 2014

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

• 대한수학회보
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• 제34권1호
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• pp.35-42
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• 1997

Throughout this paper, all rings are assumed to be commutative with identity. By a local ring (R,m), we mean a Noetherian ring R which has a unique maximal ideal m. Let I be an ideal in a ring R and t an indeterminate over R.

• 대한수학회지
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• 제42권3호
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• pp.457-470
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• 2005

Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not difficult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we first provide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring R is weakly duo under each of the following conditions: (1) R is semilocal with nil Jacobson radical; (2) R is locally finite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring R, obtaining that i(R) is finite and $\alpha^{i(R)}R\;=\;R\alpha^{i(R)\;=\;R\alpha^{i(R)}R\;for\;all\;\alpha\;{\in}\;R$, where i(R) is the index (of nilpotency) of R. Note that one-sided Artinian rings and locally finite rings are strongly $\pi-regular$. Thus we also observe connections between strongly $\pi-regular$ weakly right duo rings and related rings, constructing available examples.

• 충청수학회지
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• 제26권4호
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• pp.799-809
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• 2013

An ideal I of a local ring (R, m, k) is said to be m-full if there exists an element $x{\in}m$ such that Im : x = I. An ideal I of a local ring R is said to have the Rees property if ${\mu}$(I) > ${\mu}$(J) for any ideal J containing I. We study properties of m-full ideals and we characterize m-primary m-full ideals in terms of the minimal number of generators of the ideals. In particular, for a m-primary ideal I of a 2-dimensional regular local ring (R, m, k), we will show that the following conditions are equivalent. 1. I is m-full 2. I has the Rees property 3. ${\mu}$(I)=o(I)+1 In this paper, let (R, m, k) be a commutative Noetherian local ring with infinite residue field k = R/m.

• 대한수학회논문집
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• 제36권1호
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• pp.1-10
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• 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.

• 대한수학회보
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• 제51권2호
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• pp.373-386
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• 2014

Let (R,m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth($M/I^nM$) = d for $n{\gg}0$. Also we show that, if dim(R) = d and $I_1{\subset}\;{\cdots}\;{\subset}I_n$ is a chain of ideals of R such that $R/I_k$ is maximal Cohen-Macaulay for all k, then $n{\leq}{\ell}_R(R/(a_1,{\ldots},a_d)R)$ for every system of parameters $a1,{\ldots},a_d$ of R. Also, in the case where dim(R) = 2, we prove that the ideal transform $D_m(R/p)$ is minimax balanced big Cohen-Macaulay, for every $p{\in}Assh_R$(R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.

• 대한수학회논문집
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• 제32권3호
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• pp.523-534
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• 2017

In this paper, we introduce two group graded types of $B{\acute{e}}zout$ modules, namely graded-$B{\acute{e}}zout$ modules and weakly graded-$B{\acute{e}}zout$ modules, which are two $B{\acute{e}}zout$ versions in Graded Module Theory. We investigate the relationship among the three types of $B{\acute{e}}zout$ modules, the ordinary $B{\acute{e}}zout$ modules and the two graded types of $B{\acute{e}}zout$ modules. Also, we study the structure of these new $B{\acute{e}}zout$ modules along with different properties; for instance, "A graded-$B{\acute{e}}zout$ R-module, with R being a Noetherian ring, is Noetherien iff it is gr-Noetherian".

• 대한수학회보
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• 제55권1호
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• pp.311-317
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• 2018

Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module $Hom_R(R/I,H^t_I(R))$ is finitely generated, then $$t={\sup}\{{\dim}{\widehat{\hat{R}_p}}/Q:p{\in}V(I{\hat{R}}),\;Q{\in}mAss{_{\widehat{\hat{R}_p}}}{\widehat{\hat{R}_p}}\;and\;p{\widehat{\hat{R}_p}}=Rad(I{\wideha{\hat{R}_p}}=Q)\}$$. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R{\subset}R[X]$ will be included.