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Bases of minimal elements of some partially ordered free Abelian groups. (English) Zbl 1101.16010

A subsemigroup \(A\) of \(\mathbb{N}_0^k\) is called full affine if for any \(a\in A\) and \(b\in\mathbb{N}_0^k\) the element \(b\) lies in \(A\) whenever \(a+b\in A\). A partial order on an Abelian group \(G\) is called unperforated if for all \(g\in G\) and all positive integers \(n\), if \(ng\in G^+\) then \(g\in G^+\).
Let \(G\) be a directed unperforated partially ordered Abelian group such that \(G^+\) is a finitely generated semigroup. Then \(G\) is free and has a free basis of minimal elements of \(G^+\) (Theorem 2.8).
If \(R\) is a semilocal ring then there are non-zero finitely generated indecomposable projective \(R\)-modules, say \(P_1,\dots,P_k\), such that for any finitely generated projective module \(Q\) there exist unique numbers \(n_1,\dots,n_k,m_1,\dots,m_k\in\mathbb{N}_0\) such that \(n_im_i=0\) for \(i=1,\dots,k\), and \(Q\oplus P_1^{(n_1)}\oplus\cdots\oplus P_k^{(n_k)}\cong P_1^{(m_1)}\oplus\cdots\oplus P_k^{(m_k)}\) (Theorem 3.2).

MSC:

16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
16E20 Grothendieck groups, \(K\)-theory, etc.
16D40 Free, projective, and flat modules and ideals in associative algebras
20M14 Commutative semigroups
20F60 Ordered groups (group-theoretic aspects)
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