Congruences in \({\mathbb{Z}}^ n\), finite Abelian groups and the Chinese remainder theorem
Let \({\mathbb{Z}}^ n\) denote the additive group of column n-vectors with integral coordinates and \({\mathbb{Z}}^{n\times n}\) the ring of \(n\times n\) integral matrices. If \(M\in {\mathbb{Z}}^{n\times n}\) is a regular matrix, then the author generalizes the concept of the congruence to \({\mathbb{Z}}^ n\) in saying that \({\mathfrak a}\equiv {\mathfrak b}\) (mod M) for \({\mathfrak a}\equiv {\mathfrak b}\in {\mathbb{Z}}^ n\) if the difference \({\mathfrak a}-{\mathfrak b}\) belongs to \(M{\mathbb{Z}}^ n\), the set of all linear integral combinations of column vectors of M. He shows, using the Smith normal form theorem for M, that every finite Abelian group G is realizable as the quotient group \({\mathbb{Z}}^ n/M{\mathbb{Z}}^ n\) of integral vectors modulo M. As a by-product a simple proof of the Chinese remainder theorem is given.
scientific article; zbMATH DE number 4025504