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Axioms for invariant factors. (English) Zbl 0887.15003

Recall that an integral domain \(R\) is called an elementary divisor domain [cf. I. Kaplansky, Trans. Am. Math. Soc. 66, 464-491 (1949; Zbl 0036.01903)] if every matrix \(A\) over \(R\) is equivalent to a Smith normal form, i.e. there exist matrices \(U\), \(V\) over \(R\), both invertible satisfying \[ UAV= \text{diag} \bigl(s_1(A), s_2(A), \dots \bigr), \] where \(s_i (A)\) divides \(s_{i+1}(A)\), are invariant factors of \(A\), uniquely determined up to units of \(R\) by \(A\), where \(s_k(A)= d_k(A)/d_{k-1} (A)\), \(k=1, \dots, \text{rank}(A)\), and \(d_k (A)=\) g.c.d. of all \(k\times k\) minors of \(A\).
The author proves that \(s_k\) is uniquely determined by its suitably chosen five properties.

MSC:

15A21 Canonical forms, reductions, classification

Citations:

Zbl 0036.01903
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