• 대한수학회논문집
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• 제38권4호
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• pp.993-999
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• 2023

Let A be a G-graded commutative ring with identity and M a graded A-module. Let m, n be positive integers with m > n. A proper graded submodule L of M is said to be graded (m, n)-closed if a^m_g·x_t ∈ L implies that aⁿ_g·x_t ∈ L, where a_g ∈ h(A) and x_t ∈ h(M). The aim of this paper is to explore some basic properties of these class of submodules which are a generalization of graded (m, n)-closed ideals. Also, we investigate GC^m_n - rad property for graded submodules.

• 대한수학회논문집
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• 제38권4호
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• pp.1001-1017
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• 2023

Let R be a commutative graded ring with nonzero identity and n a positive integer. Our principal aim in this paper is to introduce and study the notions of graded n-irreducible and strongly graded n-irreducible ideals which are generalizations of n-irreducible and strongly n-irreducible ideals to the context of graded rings, respectively. A proper graded ideal I of R is called graded n-irreducible (respectively, strongly graded n-irreducible) if for each graded ideals I₁, . . . , I_n+1 of R, I = I₁ ∩ · · · ∩ I_n+1 (respectively, I₁ ∩ · · · ∩ I_n+1 ⊆ I ) implies that there are n of the I_i 's whose intersection is I (respectively, whose intersection is in I). In order to give a graded study to this notions, we give the graded version of several other results, some of them are well known. Finally, as a special result, we give an example of a graded n-irreducible ideal which is not an n-irreducible ideal and an example of a graded ideal which is graded n-irreducible, but not graded (n - 1)-irreducible.

• 대한수학회지
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• 제39권1호
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• pp.127-135
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• 2002

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( $a_1$, $a_2$,ㆍㆍㆍ, $a_{s}$ ) generated by s elements in A and G(I) =(equation omitted)$_{n 0}$ $I^{n}$ / $I^{n＋1}$ of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h $t_{A}$ I and a(G(I)) denotes the a-invariant of G(I). Let $X_{A}$$^{s}$ denote the set of equimultiple good ideals I in A with h $t_{A}$ I = s, R(I) = A [It] be the Rees algebra of I, and $K_{R(I)}$ denote the canonical module of R(I). Let a I such that $I^{n＋l}$ = a $I^{n}$ for some n$\geq$0 and $\mu$$_{A}$(I)$\geq$2, where $\mu$$_{A}$(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) $K_{R(I)}$(equation omitted)R(I)＋as graded R(I)-modules. (2) $I^2$ = aI and aA : I$\in$ $X^1$$_{A}$._{A}$./.

• Advances in concrete construction
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• 제9권6호
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• pp.557-568
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• 2020

In this study, the impact of ring supports around the shell circumferential has been examined for their various positions along the shell axial length using Rayleigh-Ritz formulation. These shells are stiffened by rings in the tangential direction. For isotropic materials, the physical properties are same everywhere where the laminated and functionally graded materials, they vary from point to point. Here the shell material has been taken as functionally graded material. The influence of the ring supports is investigated at various positions. These variations have been plotted against the locations of ring supports for three values of length-to-diameter ratios. Effect of ring supports with middle layer thickness is presented using the Rayleigh-Ritz procedure with three different conditions. The influence of the positions of ring supports for clamped-clamped is more visible than simply supported and clamped-free end conditions. The frequency first increases and gain maximum value in the midway of the shell length and then lowers down. The Lagrangian functional is created by adding the energy expressions for the shell and rings. The axial modal deformations are approximated by making use of the beam functions. The comparisons of frequencies have been made for efficiency and robustness for the present numerical procedure. Throughout the computation, it is observed that the frequency behavior for the boundary conditions follow as; clamped-clamped, simply supported-simply supported frequency curves are higher than that of clamped-simply curves. To generate the fundamental natural frequencies and for better accuracy and effectiveness, the computer software MATLAB is used.

• 대한수학회논문집
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• 제11권2호
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• pp.295-301
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• 1996

In this paper we will define correct module and strongly correct module. We can have some basic results about those modules. And we will show that M is a graded correct R-module if and only if $M_e$ is a correct $R_e$-module.

• 호남수학학술지
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• 제29권4호
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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.

• Advances in nano research
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• 제12권3호
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• pp.337-344
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• 2022

In this work, the vibrational frequency of two layered FGM cylindrical shell with and without the effects of internal pressure under ring support are discussed in detailed. The functionally graded materials of a cylindrical shell are designed for specific purpose and studied under various boundary conditions. The Love shell dynamical equations theory is utilized to find the relationship between the curvature displacement and strain displacement. Natural frequency vibrations are analyzed by using volume polynomial for bi-layered FGM shell under ring support both for with and without internal pressures.

• 대한수학회지
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• 제49권2호
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• pp.435-447
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• 2012

Let G be a group. Let R be a G-graded commutative ring with identity and M be a G-graded multiplication module over R. A proper graded submodule Q of M is semiprime if whenever $I^nK{\subseteq}Q$, where $I{\subseteq}h(R)$, n is a positive integer, and $K{\subseteq}h(M)$, then $IK{\subseteq}Q$. We characterize semiprime submodules of M. For example, we show that a proper graded submodule Q of M is semiprime if and only if grad$(Q){\cap}h(M)=Q+{\cap}h(M)$. Furthermore if M is finitely generated then we prove that every proper graded submodule of M is contained in a graded semiprime submodule of M. A proper graded submodule Q of M is said to be almost semiprime if (grad(Q)$\cap$h(M))n(grad$(0_M){\cap}h(M)$) = (Q$\cap$h(M))n(grad$(0_M){\cap}Q{\cap}h(M)$). Let K, Q be graded submodules of M. If K and Q are almost semiprime in M such that Q + K $\neq$ M and $Q{\cap}K{\subseteq}M_g$ for all $g{\in}G$, then we prove that Q + K is almost semiprime in M.

• 대한수학회지
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• 제43권4호
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• pp.871-883
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• 2006

We show that the Auslander index is the same as the column invariant over Gorenstein local rings. We also show that Ding's conjecture ([13]) holds for an isolated non-Gorenstein ring A satisfying a certain condition which seems to be weaker than the condition that the associated graded ring of A is Cohen-Macaulay.

• 대한수학회지
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• 제54권6호
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• pp.1733-1757
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• 2017

Let $R={\oplus}_{{\alpha}{\in}{\Gamma}}R_{\alpha}$ be an integral domain graded by an arbitrary torsionless grading monoid ${\Gamma}$, ${\bar{R}}$ be the integral closure of R, H be the set of nonzero homogeneous elements of R, C(f) be the fractional ideal of R generated by the homogeneous components of $f{\in}R_H$, and $N(H)=\{f{\in}R{\mid}C(f)_v=R\}$. Let $R_H$ be a UFD. We say that a nonzero prime ideal Q of R is an upper to zero in R if $Q=fR_H{\cap}R$ for some $f{\in}R$ and that R is a graded UMT-domain if each upper to zero in R is a maximal t-ideal. In this paper, we study several ring-theoretic properties of graded UMT-domains. Among other things, we prove that if R has a unit of nonzero degree, then R is a graded UMT-domain if and only if every prime ideal of $R_{N(H)}$ is extended from a homogeneous ideal of R, if and only if ${\bar{R}}_{H{\backslash}Q}$ is a graded-$Pr{\ddot{u}}fer$ domain for all homogeneous maximal t-ideals Q of R, if and only if ${\bar{R}}_{N(H)}$ is a $Pr{\ddot{u}}fer$ domain, if and only if R is a UMT-domain.