• The Pure and Applied Mathematics
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• v.14 no.3
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• pp.223-230
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• 2007

We give an answer to the following question: Question. Let S be a subset of [0,1] containing a maximal element m > 0 and let C :=$\{I_{t}\;{\mid}\;t{\in}S\}$ be a decreasing chain of ideals of a BCK/BCI-algebra X. Then does there exists a fuzzy ideal ${\mu}(X)=S\;and\;C_{\mu}=C?$.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1505-1519
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• 2017

Assume that M is an R-module where R is a commutative ring. A proper submodule N of M is called a weakly 2-absorbing primary submodule of M if $0{\neq}abm{\in}N$ for any $a,b{\in}R$ and $m{\in}M$, then $ab{\in}(N:M)$ or $am{\in}M-rad(N)$ or $bm{\in}M-rad(N)$. In this paper, we extended the concept of weakly 2-absorbing primary ideals of commutative rings to weakly 2-absorbing primary submodules of modules. Among many results, we show that if N is a weakly 2-absorbing primary submodule of M and it satisfies certain condition $0{\neq}I_1I_2K{\subseteq}N$ for some ideals $I_1$, $I_2$ of R and submodule K of M, then $I_1I_2{\subseteq}(N:M)$ or $I_1K{\subseteq}M-rad(N)$ or $I_2K{\subseteq}M-rad(N)$.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.199-209
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• 2005

An ideal space is ${\cal{I}}-resolvable$ if it has two disjoint ${\cal{I}}-dense$ subsets. We answer the question: If X is ${\cal{I}}-resolvable$, then is X (${\cal{I}}\;{\cup}\;{\cal{N}$)-resolvable?, posed by Dontchev, Ganster and Rose. We give three generalizations of the well known Banach Category Theorem and deduce the Banach category Theorem as a corollary. Characterizations of completely codense ideals and ${\cal{I}-locally$ closed sets are given and their properties are discussed.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.715-736
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• 2012

Brown provided a structure theorem for a class of perfect ideals of grade 3 with type ${\lambda}$ > 0. We introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4 in a Noetherian local ring. We construct a class of perfect ideals I of grade 3 with type 2 defined by a certain skew-symmetrizable matrix. We present the Hilbert function of the standard $k$-algebras R/I, where R is the polynomial ring $R=k[v_0,v_1,{\ldots},v_m]$ over a field $k$ with indeterminates $v_i$ and deg $v_i=1$.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.983-992
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• 2023

Let A be a ring and 𝓙 = {ideals I of A | J(I) = I}. The Krull dimension of A, written dim A, is the sup of the lengths of chains of prime ideals of A; whereas the dimension of the maximal spectrum, denoted by dim _𝓙A, is the sup of the lengths of chains of prime ideals from 𝓙. Then dim _𝓙A ≤ dim A. In this paper, we will study the dimension of the maximal spectrum of some constructions of rings and we will be interested in the transfer of the property J-Noetherian to ring extensions.

• Kyungpook Mathematical Journal
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• v.27 no.2
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• pp.181-186
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• 1987

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.251-258
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• 2002

Let R be a commutative Noetherian ring and M an R-module. In this paper we will consider the asymptotic behaviour of ideals relative to an R-module E which is M-injective.

• Journal of applied mathematics & informatics
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• v.5 no.1
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• pp.99-110
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• 1998

In this paper we study cyclic codes in $Z_m$. i.e., ideals in $Z_mG$, G afinite abelian group and we give a classification of such codes. We also sgtudy the minimum Hamming distance and the generalized Hamming weight of BCH codes over $Z_m$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1341-1352
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• 2016

Let R be a commutative Noetherian ring, ${\Phi}$ a system of ideals of R and $I{\in}{\Phi}$. In this paper among other things we prove that if M is finitely generated and $t{\in}\mathbb{N}$ such that the R-module $H^i_{\Phi}(M)$ is $FD_{{\leq}1}$ (or weakly Laskerian) for all i < t, then $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all i < t and for any $FD_{{\leq}0}$ (or minimax) submodule N of $H^t_{\Phi}(M)$, the R-modules $Hom_R(R/I,H^t_{\Phi}(M)/N)$ and $Ext^1_R(R/I,H^t_{\Phi}(M)/N)$ are finitely generated. Also it is shown that if cd I = 1 or $dimM/IM{\leq}1$ (e.g., $dim\;R/I{\leq}1$) for all $I{\in}{\Phi}$, then the local cohomology module $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all $i{\geq}0$. These generalize the main results of Aghapournahr and Bahmanpour [2], Bahmanpour and Naghipour [6, 7]. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.717-723
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• 2012

In this paper, we investigate when gr-multiplication modules are finitely generated and show that if M is a finitely generated gr-multiplication R-module then there is a lattice isomorphism between the lattice of all graded ideals I of R containing ann(M) and the lattice of all graded submodules of M.