• 대한수학회보
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• 제32권2호
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• pp.321-328
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• 1995

In this note all are commutative rings with identity and all modules are unital. Let R be a ring. An R-module M is called a multiplication module if for every submodule N of M there esists an ideal I of R such that N = IM. Clearly the ring R is a multiplication module as a module over itself. Also, it is well known that invertible and more generally profective ideals of R are multiplication R-modules (see [11, Theorem 1]).

• 대한수학회논문집
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• 제15권2호
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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).

• 대한수학회보
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• 제31권2호
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• pp.163-165
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• 1994

Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.

• 대한수학회지
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• 제58권6호
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• pp.1513-1528
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• 2021

Let R be a commutative ring with prime nilradical Nil(R) and M an R-module. Define the map 𝜙 : R → R_Nil(R) by ${\phi}(r)=\frac{r}{1}$ for r ∈ R and 𝜓 : M → M_Nil(R) by ${\psi}(x)=\frac{x}{1}$ for x ∈ M. Then 𝜓(M) is a 𝜙(R)-module. An R-module P is said to be 𝜙-projective if 𝜓(P) is projective as a 𝜙(R)-module. In this paper, 𝜙-exact sequences and 𝜙-projective R-modules are introduced and the rings over which all R-modules are 𝜙-projective are investigated.

• 대한수학회지
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• 제51권6호
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• pp.1305-1319
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• 2014

A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism $N{\rightarrow}S$ can be extended to a homomorphism $M{\rightarrow}S$. M is called coneat-flat if the kernel of any epimorphism $Y{\rightarrow}M{\rightarrow}0$ is coneat in Y. It is proven that (1) coneat submodules of any right R-module are coclosed if and only if R is right K-ring; (2) every right R-module is coneat-flat if and only if R is right V -ring; (3) coneat submodules of right injective modules are exactly the modules which have no maximal submodules if and only if R is right small ring. If R is commutative, then a module M is coneat-flat if and only if $M^+$ is m-injective. Every maximal left ideal of R is finitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring R is perfect if and only if every coneat-flat module is projective. We also study the rings over which coneat-flat and flat modules coincide.

• Kyungpook Mathematical Journal
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• 제49권2호
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• pp.255-263
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• 2009

For an endomorphism ${\alpha}$ of R, in [1], a module $M_R$ is called ${\alpha}$-compatible if, for any $m{\in}M$ and $a{\in}R$, ma = 0 iff $m{\alpha}(a)$ = 0, which are a generalization of ${\alpha}$-reduced modules. We study on the relationship between the quasi-Baerness and p.q.-Baer property of a module MR and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [2] and some results in [9]. In particular, we show: for an ${\alpha}$-compatible module $M_R$ (1) $M_R$ is p.q.-Baer module iff $M[x;{\alpha}]_{R[x;{\alpha}]}$ is p.q.-Baer module. (2) for an automorphism ${\alpha}$ of R, $M_R$ is p.q.-Baer module iff $M[x,x^{-1};{\alpha}]_{R[x,x^{-1};{\alpha}]}$ is p.q.-Baer module.

• 대한수학회보
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• 제59권5호
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• pp.1289-1304
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• 2022

In this note, we show that a strongly 𝜙-ring R is a 𝜙-PvMR if and only if any 𝜙-torsion-free R-module is 𝜙-w-flat, if and only if any GV-torsion-free divisible R-module is nonnil-absolutely w-pure, if and only if any GV-torsion-free h-divisible R-module is nonnil-absolutely w-pure, if and only if any finitely generated nonnil ideal of R is w-projective.

• 항공우주기술
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• 제11권2호
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• pp.1-11
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• 2012

본 논문은 우주용 X-대역 능동 위상 배열 레이더를 위한 T/R 모듈에 대하여 기술한다. T/R 모듈은 위상 배열 안테나의 전자적 빔 조향을 가능하게 하는 송수신 RF 반도체 소자가 집적된 모듈이며 SAR 탑재체의 핵심 부품이다. 기존의 T/R 모듈은 수평 편파 또는 수직 편파 신호를 스위치에 의해 선택적으로 수신할 수 있었다. 그러나 현재 기술적 흐름은 수평 편파와 수직 편파 신호를 동시에 수신하는 것이다. 따라서 이중 편파 동시 수신 T/R 모듈에 대한 개발이 활발히 이루어지고 있다. 본 연구는 차세대 SAR 탑재체를 위한 선행 연구로서 능동 위상 배열 레이더 T/R 모듈의 기술 개발 동향과 이중 수신 T/R 모듈의 예비 설계 결과를 정리하였다.

• East Asian mathematical journal
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• 제16권1호
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• pp.111-123
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• 2000

Northcott and McKerrow proved that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-1}]$ is an injective left R[xl-module. Park generalize Northcott and McKerrow's result so that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-S}]$ is an injective left $R[x^s]$-module, where S is a submonoid of N(N is the set of all natural numbers). In this paper we show $$Hom_{R[x^S]}(M[x^{-S}],\;N[x^{-S}]){\cong}Hom_R(M,\;N)[[x^S]]$$ and using the above result and this isomorphism, finally we show that $$Ext^i_{R[x^S]}(M[x^{-S}],\;N[x^{-S}]){\cong}Ext^i_R(M,\;N)[[x^S]]$$.

• Kyungpook Mathematical Journal
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• 제48권1호
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• pp.143-154
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• 2008

Lomp [9] has studied finitely generated projective modules over semilocal rings. He obtained the following: finitely generated projective modules over semilocal rings are semilocal. We shall give necessary and sufficient conditions for finitely generated modules to be semilocal modules. By using a lifting property, we also give characterizations of right perfect (semiperfect) rings. Our main results can be summarized as follows: (1) Let M be a finitely generated module. Then M has finite hollow dimension if and only if M is weakly supplemented if and only if M is semilocal. (2) A ring R is right perfect if and only if every flat right R-module is lifting and every right R-module has a flat cover if and only if every quasi-projective right R-module is lifting. (3) A ring R is semiperfect if and only if every finitely generated flat right R-module is lifting if and only if RR satisfies the lifting property for simple factor modules.