Torsion-Free and Divisible Modules Over Non-Integral-Domains

Levy, Lawrence S.

doi:10.4153/cjm-1963-016-1

Cited by 139 publications

(45 citation statements)

References 9 publications

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“…Moreover, R is of finite right rank and, hence, every finitely generated nonsingular right fi-module can be embedded in a free module [4,Corollary XII,7.3]. Since every finitely generated right torsion-free fi-module is right nonsingular, it follows by Levy's theorem [2,Theorem 5.3] that R is right Ore. By [1,Theorem 6.25], Theorem 1, and [1, Theorem 6.26], R is left PCI. □…”

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confidence: 81%

A Right PCI Ring is Right Noetherian

Damiano¹

1979

Proceedings of the American Mathematical Society

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Abstract. C. Faith and J. Cozzens have shown that a ring, whose right proper cyclic modules are injective, is either semisimple or a simple, right semihereditary, right Ore K-domain. They have posed a question as to whether such a ring is right noetherian. In this paper, an affirmative answer is given to that question. Moreover, necessary and sufficient conditions are given as to when a right PCI ring is left PCI.In [1], Faith and Cozzens proved that a ring R whose proper right cyclic modules are injective must be either semisimple or a simple, right semihereditary, right Ore F-domain. They noted that all the known examples of such rings are right noetherian and posed the question whether every ring with this property is noetherian. We will answer this question in the affirmative. More clearly, we will show: Theorem 1. Let R be a right PCI ring, then either

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confidence: 81%

A Right PCI Ring is Right Noetherian

Damiano¹

1979

Proceedings of the American Mathematical Society

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“…It follows that Hom R (Q, U ) = 0 for every uniform right ideal U of R. Therefore every uniform right ideal of R contains a nonzero divisible submodule. It is well known that in a semiprime right Goldie ring S, every nonsingular divisible S-module is injective, see instance [9,Theorem 3.3]. Hence every uniform right ideal of R contains a nonzero injective submodule.…”

Section: Proof By Proposition 21(d) Each (R/i I )mentioning

confidence: 99%

On Subdirect Product of Prime Modules

Dehghani¹,

Vedadi²

2017

Communications of the Korean Mathematical Society

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Abstract. In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

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“…For the proof of the necessary condition, let E be a torsion-free and divisible left ^4-module and consider the diagram (May o->a->. 4 * I E where Ct is a left ideal of A. Let aEQ, and consider</>(a).…”

Section: Theorem 1 If a Is A Left Ore Domain Then A Torsion-free Lementioning

confidence: 99%