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The ideal of relations of a nilpotent diagonal matrix. (English) Zbl 0902.15007

Let \(R\) be a commutative ring, and let \(M_n(R)\) be the \(R\)-algebra of \(n\times n\) matrices over \(R\). Let \(A\) be an element of \(M_n(R)\). Then the ideal of relations satisfied by \(A\), and denoted \(I(A)\), is defined by the set of polynomials in one variable, \(f\) with coefficients in \(R\) such that \(f(A)= 0\) in \(M_n(R)\). The authors give an explicit description of the ideal of relations when \(A\) is a nilpotent diagonal matrix. However, to the best of the knowledge of the present reviewer, a characterization of the ideal of relations for an arbitrary matrix over any commutative ring has been given by Neal H. McCoy [Bull. Am. Math. Soc. 45, 280-284 (1939; Zbl 0021.00404)] and we find more details in the book of B. R. McDonald [Linear algebra over commutative rings (1984; Zbl 0556.13003)]. The paper is selfcontained.

MSC:

15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A30 Algebraic systems of matrices
15A24 Matrix equations and identities
16S50 Endomorphism rings; matrix rings
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