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\(K_0\) of a semilocal ring. (English) Zbl 0955.13006

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.
Reviewer: T.Porter (Bangor)

MSC:

13D15 Grothendieck groups, \(K\)-theory and commutative rings
19A49 \(K_0\) of other rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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