## 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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