• 대한수학회논문집
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• 제28권4호
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• pp.669-677
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• 2013

In this paper we introduce the notion of NI near-rings similar to the notion introduced in rings. We give topological properties of collection of strongly prime ideals in NI near-rings. We have shown that if N is a NI and weakly pm near-ring, then $Max(N)$ is a compact Hausdorff space. We have also shown that if N is a NI near-ring, then for every $a{\in}N$, $cl(D(a))=V(N^*(N)_a)=Supp(a)=SSpec(N){\setminus}int\;V(a)$.

• East Asian mathematical journal
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• 제17권2호
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• pp.275-286
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• 2001

In this paper, a skew polynomial ring $R[x;\alpha]$ of a ring R with a monomorphism $\alpha$ are investigated as follows: For a reduced ring R, assume that $\alpha(P){\subseteq}P$ for any minimal prime ideal P in R. Then (i) $R[x;\alpha]$ is a reduced ring, (ii) a ring R is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring) if and only if the skew polynomial ring $R[x;\alpha]$ is Baer(resp. quasi-Baer, p.q.-Baer, a p.p.-ring).

• 대한수학회보
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• 제59권3호
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• pp.529-545
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• 2022

This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring R is called right CIFD if R/I is right duo by some proper ideal I of R such that I is contained in the center of R. We first see that this property is seated between right duo and right π-duo, and not left-right symmetric. We prove, for a right CIFD ring R, that W(R) coincides with the set of all nilpotent elements of R; that R/P is a right duo domain for every minimal prime ideal P of R; that R/W(R) is strongly right bounded; and that every prime ideal of R is maximal if and only if R/W(R) is strongly regular, where W(R) is the Wedderburn radical of R. It is also proved that a ring R is commutative if and only if D₃(R) is right CIFD, where D₃(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring R is right CIFD if and only if R/I is commutative by a proper ideal I of R contained in the center of R.

• 대한수학회보
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• 제40권2호
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• pp.223-234
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• 2003

This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $_{A}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $_{A}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).

• 대한수학회지
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• 제57권3호
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• pp.571-583
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• 2020

The Galois ring R of characteristic pⁿ having p^mn elements is a finite extension of the ring of integers modulo pⁿ, where p is a prime number and n, m are positive integers. In this paper, we develop the concepts of Jacobi sums over R and under the assumption that the generating additive character of R is trivial on maximal ideal of R, we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.

• 대한수학회논문집
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• 제20권4호
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• pp.645-648
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• 2005

Let K be a Galois extension of a field F with G = Gal(K/F). Let L be an extension of F such that $K\;{\otimes}_F\;L\;=\; N_1\;{\oplus}N_2\;{\oplus}{\cdots}{\oplus}N_k$ with corresponding primitive idempotents $e_1,\;e_2,{\cdots},e_k$, where Ni's are fields. Then G acts on $\{e_1,\;e_2,{\cdots},e_k\}$ transitively and $Gal(N_1/K)\;{\cong}\;\{\sigma\;{\in}\;G\;/\;{\sigma}(e_1)\;=\;e_1\}$. And, let R be a commutative F-algebra, and let P be a prime ideal of R. Let T = $K\;{\otimes}_F\;R$, and suppose there are only finitely many prime ideals $Q_1,\;Q_2,{\cdots},Q_k$ of T with $Q_i\;{\cap}\;R\;=\;P$. Then G acts transitively on $\{Q_1,\;Q_2,{\cdots},Q_k\},\;and\;Gal(qf(T/Q_1)/qf(R/P))\;{\cong}\;\{\sigma{\in}\;G/\;{\sigma}-(Q_1)\;=\;Q_1\}$ where qf($T/Q_1$) is the quotient field of $T/Q_1$.

• 충청수학회지
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• 제27권4호
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• pp.625-633
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• 2014

We study some results which concern the types of Noetherian local rings, and improve slightly the previous result: For a complete unmixed (or quasi-unmixed) Noetherian local ring A, we prove that if either $A_p$ is Cohen-Macaulay, or $r(Ap){\leq}depth$ $A_p+1$ for every prime ideal p in A, then A is Cohen-Macaulay. Also, some analogous results for modules are considered.

• 대한수학회논문집
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• 제21권4호
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• pp.641-652
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• 2006

In [2], they found some natural generators for the ideals of the finite ring $Z_{pm}$[X]/$(X^n\;-\;1)$, where p and n are relatively prime. If p and n are not relatively prime $X^n\;-\;1$ is not a product of basic irreducible polynomials but a product of primary polynomials. Due to this fact, to consider the ideals of $Z_{pm}$[X]/$(X^n\;-\;1)$ in 'inseparable' case we need to look at the primary ideals of $Z_{pm}$[X]. In this paper, we find a set of generators of ideals of $Z_{pm}$[X]/(f) for some primary polynomials f of $Z_{pm}$[X].

• 대한수학회지
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• 제50권5호
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• pp.991-1008
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• 2013

In this paper, we mainly discuss Gorenstein Dedekind do-mains (G-Dedekind domains for short) and their overrings. Let R be a one-dimensional Noetherian domain with quotient field K and integral closure T. Then it is proved that R is a G-Dedekind domain if and only if for any prime ideal P of R which contains ($R\;:_K\;T$), P is Gorenstein projective. We also give not only an example to show that G-Dedekind domains are not necessarily Noetherian Warfield domains, but also a definition for a special kind of domain: a 2-DVR. As an application, we prove that a Noetherian domain R is a Warfield domain if and only if for any maximal ideal M of R, $R_M$ is a 2-DVR.

• 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).