\(K_0\) of a semilocal ring
A semilocal ring \(R\) is one for which \(R/J(R)\) is artinian, \(J(R)\) being the Jacobson radical. Studying direct sum decomposition of modules having semilocal endomorphism rings, leads naturally to consideration of \(K_0(R)\) for \(R\) semilocal. As \(R/J(R)\) is artinian, \(K_0(R/J(R))\) is a partially ordered abelian group with order unit, which is isomorphic to \((\mathbb{Z}^n, \leq,U)\) where \(\leq \) is componentwise and \(U\) is an order unit in \((\mathbb{Z}^n, \leq)\). The group \(K_0(R)\) is also a partially ordered group and the natural quotient map \(\pi:R\to R/J(R)\) induces an order embedding of \(K_0(R)\) into \(K_0(R/J(R))\). The failure of the Krull-Schmidt theorem for finitely generated projective modules over some semilocal rings shows that the embedding is sometimes onto a proper ordered subgroup. This raises the natural question as to which proper ordered subgroups of \((\mathbb{Z}^n,\leq)\) can arise in this way. The answer given is that any such subgroup can so arise.
scientific article; zbMATH DE number 1423603
scientific article