Matrices over rings of algebraic integers (Q807716)

For matrices over a domain R of algebraic integers the authors investigate: (i) completions, (ii) the Hermite form, (iii) the Schur form, (iv) the Smith form, and (v) links relating eigenvalues and Smith invariants. These items have been investigated previously but not in the present context. The results in this paper are given in terms of a domain of algebraic integers \(R'\) which contains the base ring R and with the property that every ideal in R becomes a principal ideal when extended to \(R'\) (even though neither R nor \(R'\) is necessarily a principal ideal ring). An example of the kind of results obtained is the following theorem: Let A be an \(n\times n\) matrix over R. Then for a suitable extension \(R'\) of R there exist unimodular matrices \(U'\) and \(V'\) over \(R'\) such that \(U'AV'=Diag[s_ 1',s_ 2',...]\) where the Smith invariants \(s_ 1',s_ 2',..\). are in \(R'\) and, of course, for each i \(s_ i'\) is a divisor of \(s'_{i+1}\).