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On products of singular elements. (English) Zbl 0788.16027

Some rings, like the ring \(M(n,K)\) of square matrices, do not contain irreducible elements: any singular element \(x\) can be written as the product \(x = yz\) of two singular elements \(y\) and \(z\). These rings are called \(S\)-rings. The first purpose of this paper is to exhibit some examples of \(S\)-rings. In Part 2, the authors study in an elementary way the ring \(K[A]\) by giving a necessary and sufficient condition in order that the matrix \(A\) could be written as \(P(A)Q(A)\), where \(P\) and \(Q\) are polynomials, with \(P(A)\) and \(Q(A)\) two singular matrices. Part 4 is devoted to the solution of the following non trivial problem: given any matrix \(A\), what is the maximal number \(n(A)\) of singular and permutable matrices \(A_ i\) such that \(A = A_ 1\cdots A_ m\)? A simple observation allows us to answer the same problem, for \(A\) and \(A_ i\) bistochastic.
Reviewer: Y.Kuo (Knoxville)

MSC:

16U99 Conditions on elements
15A54 Matrices over function rings in one or more variables
15A24 Matrix equations and identities
16S50 Endomorphism rings; matrix rings
11C20 Matrices, determinants in number theory
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References:

[1] Glazman-Liubitch, Analyse linéaire dans les espaces de dimension finie, Editions Mir, Moscou (1972). · Zbl 0243.15002
[2] Jacobson, N., Lectures in abstract algebra II Linear algebra, D. Van NostrandNew-York, (1953). · Zbl 0053.21204
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