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The flat dimension of mixed abelian groups as \(E\)-modules. (English) Zbl 0843.20045
Let \(G\) be an abelian group and let \(E = E(G)\) denote its endomorphism ring. Then \(G\) is in a natural way a left \(E\)-module. It was R. S. Pierce who asked for a way to calculate the flat dimension \(fd_E(G)\) of the \(E\)-module \(_EG\). The present paper is devoted to this topic for the case that \(G\) belongs to a certain class \(\mathcal G\) of mixed abelian groups which was introduced by S. Glaz and W. Wickless [Commun. Algebra 22, No. 4, 1161-1176 (1994; Zbl 0801.20037)]. Let \(G \in {\mathcal G}\) and let \(T\) denote the maximal divisible subgroup of \(G\). Then \(G/T\) is a vector space over the field \(\mathbb{Q}\) of rational numbers of some finite dimension \(n\) which gives rise to a ring homomorphism \(\mu\) from \(E(G)\) into the ring \(M_n(\mathbb{Q})\) of \(n \times n\) rational matrices. Let \(A\) denote the image of \(\mu\). Then \(A\) is a subalgebra of \(M_n(\mathbb{Q})\) which is unique up to isomorphism and is said to be realized by \(G\). Both \(G/T\) and \(M_n(\mathbb{Q})\) may be regarded as \(A\)-modules. The authors show that, for any \(G \in {\mathcal G}\), \(fd_E(G) = fd_A (G/T)\) and relate the projective dimension of the \(E\)-module \(G\) to the flat dimension of the \(A\)-module \(G/T\). They prove that, given any \(k\) with \(0 \leq k \leq \infty\), there exists \(G \in {\mathcal G}\) such that \(fd_E(G) = k\). The authors consider the question which subalgebras of \(M_n(\mathbb{Q})\) are realizable by some \(G \in {\mathcal G}\) and show that this class is fairly large. An example is given of a subalgebra of \(M_4(\mathbb{Q})\) which is not realizable.
Reviewer: J.Hausen (Houston)

20K21 Mixed groups
16E10 Homological dimension in associative algebras
16S50 Endomorphism rings; matrix rings
16D40 Free, projective, and flat modules and ideals in associative algebras
20K20 Torsion-free groups, infinite rank
Full Text: DOI
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