• 대한수학회논문집
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• 제20권2호
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• pp.275-290
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• 2005

Let $S\;=\;R[\chi_{ij}\mid1\;{\le}\;i\;{\le}\;m,\;1\;{\le}\;j\;{\le}\;n]$ be the polynomial ring over a noetherian commutative ring R and $I_p$ be the determinantal ideal generated by the $p\;\times\;p$ minors of the generic matrix $(\chi_{ij})(1{\le}P{\le}min(m,n))$. We describe a minimal free resolution of $S/I_{p}$, in the case m = n = p + 2 over $\mathbb{Z}$.

• 대한수학회논문집
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• 제38권3호
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• pp.705-714
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• 2023

Bennis and El Hajoui have defined a (commutative unital) ring R to be S-coherent if each finitely generated ideal of R is a S-finitely presented R-module. Any coherent ring is an S-coherent ring. Several examples of S-coherent rings that are not coherent rings are obtained as byproducts of our study of the transfer of the S-coherent property to trivial ring extensions and amalgamated duplications.

• 대한수학회논문집
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• 제39권1호
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• pp.45-58
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• 2024

Let R be a commutative ring with identity. If the nilpotent radical N il(R) of R is a divided prime ideal, then R is called a ϕ-ring. Let R be a ϕ-ring and S be a multiplicative subset of R. In this paper, we introduce and study the class of nonnil-S-coherent rings, i.e., the rings in which all finitely generated nonnil ideals are S-finitely presented. Also, we define the concept of ϕ-S-coherent rings. Among other results, we investigate the S-version of Chase's result and Chase Theorem characterization of nonnil-coherent rings. We next study the possible transfer of the nonnil-S-coherent ring property in the amalgamated algebra along an ideal and the trivial ring extension.

• 대한수학회지
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• 제42권5호
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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).

• 대한수학회지
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• 제42권1호
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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 ${\sigma}$.

• 대한수학회지
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• 제47권3호
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• pp.633-643
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• 2010

Let a be an ideal of a commutative Noetherian ring R, M a finitely generated R-module and N a weakly Laskerian R-module. We show that if N has finite dimension d, then $Ass_R(H^d_a(N))$ consists of finitely many maximal ideals of R. Also, we find the least integer i, such that $H^i_a$(M, N) is not consisting of finitely many maximal ideals of R.

• East Asian mathematical journal
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• 제36권5호
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• pp.537-545
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• 2020

We investigate some polynomial invariants for virtual knots via virtualization moves. We define two types of polynomials W_G(t) and S²_G(t) from the definition of the index polynomial for virtual knots. And we show that they are invariants for virtual knots on the quotient ring Z[t^±1]/I where I is an ideal generated by t² - 1.

• 대한수학회지
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• 제31권4호
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• pp.619-628
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• 1994

The multiplicity theory initiated by C. Chevalley was the one with respect to ideals generated by a system of parameters of a local ring containing a field [3] and [4]. Samuel generalized the definition to primary ideals belonging the maximal ideal of a local ring which contains a field by a device which used the Hilbert characteristic function [9]. Furthermore Samuel defined multiplicity also in local rings which contain no field [10].

• 대한수학회보
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• 제45권2호
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• pp.263-267
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• 2008

Kaplansky introduced the Baer rings as rings in which every left (or right) annihilator of each subset is generated by an idempotent. On the other hand, Hattori introduced the left (resp. right) P.P. rings as rings in which every principal left (resp. right) ideal is projective. The purpose of this paper is to introduce the near-rings with Baer like condition and near-rings with P.P. like condition which are somewhat different from ring case, and to extend the results of Arendariz and Jondrup.

• 대한수학회논문집
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• 제28권1호
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• pp.39-54
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• 2013

The purpose of this paper is to find a description of the cyclic codes of length $2^n$ over $\mathbb{Z}_4$. We show that any ideal of $\mathbb{Z}_4$[X]/($X^{2n}$ - 1) is generated by at most two polynomials of the standard forms. We also find an explicit description of their duals in terms of the generators.