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On direct products of modules. (English) Zbl 0536.16025

The main result of this paper is a generalization of a theorem of S. U. Chase and may be stated as follows: Let \(G=\prod_{I}G_ i\) and \(A=\oplus_{J}A_ j\) be R-modules and let F be a filter of principal right ideals in R. Let \(f:G\to A\) be a homomorphism and denote by \(f_ j\) the composite map f followed by the projection to \(A_ j\). Let S be the ring of all subsets of I and define \[ K=\{s\in S| \quad \exists r_ sR\quad in\quad F\quad such\quad that\quad r_ sf_ j(\prod_{s}G_ i)\subseteq F(A)\quad for\quad almost\quad all\quad j\}. \] Here \(F(A)=\cap_{rR\in F}rA.\) It is shown that if \(H=<\prod_{s}G_ i| \quad s\in K>\) there is a bR in F and a subset J’ of J with finite complement such that \(bf_ j(H)\subseteq F(A)\) for all \(j\in J'\). The result of Chase reads: \(bf(\prod_{I\backslash I_ 1}G_ i)\subseteq \oplus_{J_ 1}A_ j+F(A)\) for finite subsets \(I_ 1\) and \(J_ 1\) of I and J resp. and I is countable. The author also shows that, if R is a commutative integral domain, F(A) may be replaced by D(A), the maximal divisible submodule, in Chase’s theorem and in an extension of it, proved by M. Dugas and B. Zimmermann-Huisgen [Lect. Notes Math. 874, 179-193 (1981; Zbl 0475.13011)]. It is shown that there are applications for the theory of Abelian groups.
Reviewer: L.C.A.van Leeuwen

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D80 Other classes of modules and ideals in associative algebras
20K25 Direct sums, direct products, etc. for abelian groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups

Citations:

Zbl 0475.13011
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References:

[1] Chase S.U., Pacific Journal of Math 12 pp 847– (1962) · Zbl 0115.26002 · doi:10.2140/pjm.1962.12.847
[2] Dugas M., Lecture Notes in Mathematics 874 pp 179– (1982)
[3] Fuchs L., Infinite Abelian Groups (1970) · Zbl 0209.05503
[4] Ivanov A.V., Abelian Groups and Modules pp 70– (1980)
[5] Zimmermann-Huisgen B., Arch. Math 38 pp 426– (1982) · Zbl 0495.20026 · doi:10.1007/BF01304811
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