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On direct decompositions of modules. (English. Russian original) Zbl 0804.16003

Russ. Math. 36, No. 8, 48-52 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 8(363), 52-56 (1992).
The author considers the problem of decomposability into direct sums of “small submodules” of those modules over infinite rings, whose submodules of a smaller power possess or almost possess the same property. Let \(R\) be a ring with unit, and \(\mathfrak M\) be a cardinal such that \(| R| \leq {\mathfrak M}\). The \(R\)-module \(M\) is said to be \(\mathfrak M\)-summable if \(M\) is decomposable into a direct sum of \(R\)- submodules, each being of a power not exceeding \(\mathfrak M\). The main result of the paper is the following Theorem (\(V = L\)). Let \(k\) be a non- countable regular not weakly compact cardinal, \(R\) be a ring with unit, which either is countable, or has no zero divisors, and the additive group \(R^ +\) of \(R\) be torsion-free and reduced. Also let \(\mathfrak M\) be a cardinal such that \(| R| \leq {\mathfrak M} < k\). Then there exists a family of \(R\)-modules \(\{A_ i \mid i < 2^ k\}\) (which are \(R\)- submodules of the \(R\)-module \(\prod_ k R\)) such that for any \(i < 2^ k\) the following statements are fulfilled: 1) \(| A_ i | = k\); 2) any subset \(M \subset A_ i\) of a power less than \(k\) can be embedded in a certain \(\mathfrak M\)-summable \(R\)-submodule \(M^*\) of a power less than \(k\), which is a direct summand of \(A_ i\); 3) any group homomorphism \(f_{ij} : A^ +_ i \to A^ +_ j\) for \(i \neq j\) is \(k\)-small; 4) the ring of endomorphisms \(E(A^ +_ i)\) of the group \(A^ +_ i\) is a split extension of the ring \(R\) by means of the ideal \(E_ k(A^ +_ i)\).

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
20K20 Torsion-free groups, infinite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
16S50 Endomorphism rings; matrix rings
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