• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.101-112
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• 2010

In [3] we showed that a polynomial over a Noetherian ring is divisible by some other polynomial by looking at the matrix formed by the coefficients of the polynomials which we called the resultant matrix. In this paper, we consider the polynomials with coefficients in a field and divisibility of a polynomial by a polynomial with a certain degree is equivalent to the existence of common solution to a system of Diophantine equations. As an application we construct a family of irreducible quartics over $\mathbb{Q}$ which are not of Eisenstein type.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1183-1198
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• 2013

We will introduce and study the notion of prime bases for weakly prime submodules and utilize them to derive some formulas on the weak radical of submodules of a module. In particular, we will show that every one dimensional integral domain weakly satisfies the radical formula and state some necessary conditions on local integral domains which are semi-compatible or satisfy the radical formula and also on Noetherian rings which weakly satisfy the radical formula.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.669-689
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• 2020

Cohomological degrees (or extended degrees) were introduced by Doering, Gunston and Vasconcelos as measures for the complexity of structure of finitely generated modules over a Noetherian ring. Until now only very few examples of such functions have been known. Using a Cohen-Macaulay obstruction defined earlier, we construct an infinite family of cohomological degrees.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.1017-1023
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• 2010

We prove that a maximal ideal M of D[x] has two generators and is of the form where p is an irreducible element in a PID D having infinitely many nonassociate irreducible elements and q(x) is an irreducible non-constant polynomial in D[x]. Moreover, we find how minimal generators of maximal ideals of a polynomial ring D[x] over a DVR D consist of and how many generators those maximal ideals have.

• Journal of the Chungcheong Mathematical Society
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• v.7 no.1
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• pp.153-164
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• 1994

Let A be a commutative ring with identity and M an A-module. When $U_n$ is a triangular subset of $A_n$, Sharp and Zakeri defined a module of generalized fractions $U_n^{-n}M$. In [SZ3], they described a relation of the Monomial Conjecture and a module of generalized fractions under the condition of a Noetherian local ring. In this paper, we investigate some properties of non-zero generalized fractions and give a generalization of results of Sharp and Zakeri for an arbitrary ring.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.9-16
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• 2000

An A-module M is catenary if for each pair of prime submodules K and L of M with $K{\subset}L$ all saturated chains of prime submodules of M from K to L have a common finite length. We show that when A is a Noetherian domain, then every finitely generated A-module is catenary if and only if A is a Dedekind domain or a field. Moreover, a torsion-free divisible A-module M is catenary if and only if the vector space M over Q(A) (the field of fractions of A) is finite dimensional.

• Honam Mathematical Journal
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• v.30 no.3
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• pp.513-519
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• 2008

Let (R, m) be a Noetherian local ring with maximal ideal m, E := $E_R$(R/m) and let I be an ideal of R. Let M and N be finitely generated R-modules. It is shown that $H^n_I(M,(H^n_I(N)^{\vee})){\cong}(M{\otimes}_RN)^{\vee}$ where grade(I, N) = n = $cd_i$(I, N). We also show that for n = grade(I, R), one has $End_R(H^n_I(P,R)^{\vee}){\cong}Ext^n_R(H^n_I(P,R),P^*)^{\vee}$.

• Journal of the Korean Institute of Intelligent Systems
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• v.22 no.3
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• pp.394-398
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• 2012

Based on the theory of a bipolar fuzzy set, the notion of a bipolar fuzzy subring/ideal of a Near ring is introduced and related properties are investigated. Characterizations of a bipolar fuzzy subnear ring and a bipolar fuzzy ideal in near ring are established. Relations between a bipolar fuzzy ideal and a level cut are discussed. Using bipolar fuzzy ideals, we discuss characterizations of Noetherian Near ring.

• Communications of the Korean Mathematical Society
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• v.15 no.2
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• pp.257-261
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• 2000

Northcott and Mckerrow proved that if R is a left noe-therian ring and E is an injective left R-module, then E[x-1] is an injective left R[x]-module. In this paper we generalize Northcott and McKerrow's result so that if R is a left noetherian ring and E is an in-jective left R-module, then E[x-S] is an injective left R[xS]-module, where S is a submonoid of N (N is the set of all natural numbers).

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.551-559
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• 2016

Recently a construction of local cohomology functors for compactly generated triangulated categories admitting small coproducts is introduced and studied by Benson, Iyengar, Krause, Asadollahi and their coauthors. Following their idea, we introduce the depth of objects in such triangulated categories and get that when (R, m) is a graded-commutative Noetherian local ring, the depth of every cohomologically bounded and cohomologically finite object is not larger than its dimension.