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On Modules of Singular Submodule Zero

Published online by Cambridge University Press:  20 November 2018

Vasily C. Cateforis  and
Francis L. Sandomierski
Vasily C. Cateforis
Affiliation:
University of Kentucky, Lexington, Kentucky
Francis L. Sandomierski
Affiliation:
Kent State University, Kent, Ohio
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In this paper we generalize to modules of singular submodule zero over a ring of singular ideal zero some of the results, which are well known for torsion-free modules over a commutative integral domain, e.g. [2, Chapter VII, p. 127], or over a ring, which possesses a classical right quotient ring, e.g. [13, § 5].

Let R be an associative ring with 1 and let M be a unitary right R-module, the latter fact denoted by MR. A submodule NR of MR is large in MR (MR is an essential extension of NR) if NR intersects non-trivially every non-zero submodule of MR; the notation NR ⊆′ MR is used for the statement “NR is large in MR” The singular submodule of MR, denoted Z(MR), is then defined to be the set {m ∈ M| r(m) ⊆’ RR}, where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 345 - 354
Copyright
Copyright © Canadian Mathematical Society 1971

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