an:00554446
Zbl 0806.16030
DeMeyer, Frank; Kakakhail, Haniya
Generalized equivalence of matrices over normal domains
EN
Commun. Algebra 22, No. 3, 897-904 (1994).
0092-7872 1532-4125
1994
j
16S50 13F05 20M20 13B10 20M14 15A69
homotopy classes of homomorphisms; commutative monoid; noetherian integrally closed domain; height 1 primes; epimorphism; congruence; Dedekind domain
The homotopy classes of homomorphisms of \(R\) (a commutative ring) form a commutative monoid \(M(R)\) with 0 with product. Let \(M(R)^*\) denote the non-zero elements of \(M(R)\). Let \(R\) denote a noetherian integrally closed domain, let \(X_ 1(R)\) be the set of height 1 primes of \(R\). For each \(P \in X_ 1(R)\) the localization \(R_ P\) is a discrete valuation ring.
In this article the authors examine the natural map \(\phi: M(R)^* \to \bigoplus_{P \in X_ 1(R)} M(R_ P)^*\). They show \(\phi\) is an epimorphism. They determine the congruence on \(M(R)^*\) induced by \(\phi\). As a result the authors show \(\phi\) is an isomorphism if and only if \(R\) is a Dedekind domain. They provide a unique representing matrix for each homotopy class over a Dedekind domain. These last two results improve and simplify those in Section 2 of \textit{F. R. DeMeyer} and \textit{T. J. Ford} [J. Algebra 113, 379-398 (1988; Zbl 0654.13016)].
Y.Kuo (Knoxville)
0654.13016