×

zbMATH — the first resource for mathematics

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)

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] U. Albrecht, Faithful abelian groups of infinite rank , Proc. Amer. Math. Soc. 103 (1988), 21-26. JSTOR: · Zbl 0646.20042
[2] ——–, Endomorphism rings of faithfully flat abelian groups , Res. in Math. 17 (1990), 179-201. · Zbl 0709.20031
[3] D. Arnold, Abelian groups flat over their endomorphism ring , · Zbl 0793.20051
[4] D. Arnold and C. Murley, Abelian groups, \(A\), such that \(\Hm(A,-)\) preserves direct sums of copies of \(A\) , Pacific J. Math. 56 (1975), 7-20. · Zbl 0337.13010
[5] M.R.C. Butler, On locally free torsion-free rings of finite rank , J. London Math. Soc. 43 (1968), 297-300. · Zbl 0155.07202
[6] T. Faticoni and H. Goeters, Examples of torsion-free abelian groups flat as modules over their endomorphism rings , Comm. in Alg. 19 (1991), 1-27. · Zbl 0728.20046
[7] M. Dugas and T. Faticoni, On the construction of abelian groups with prescribed endomorphism ring , · Zbl 0805.20046
[8] C. Faith, Algebra : rings, modules and categories , Vol. 1, Springer Verlag, New York, 1970. · Zbl 0266.16001
[9] L. Fuchs and K. Rangaswamy, On generalized rings , Math. Z. 107 (1968), 71-81. · Zbl 0167.03401
[10] S. Glaz and W. Wickless, Regular and principal projective endomorphism rings of mixed abelian groups , · Zbl 0801.20037
[11] R.S. Pierce, Abelian groups as modules over their endomorphism ring , Proceedings, Connecticut, 1989, University of Connecticut (1990).
[12] F. Richman and E. Walter, Primary groups as modules over their endomorphism rings , Math. Z. 89 (1965), 77-81. · Zbl 0131.25403
[13] P. Schultz, The endomorphism ring of the additive group of a ring , J. Austral. Math. Soc. 15 (1973), 60-69. · Zbl 0257.20037
[14] B. Stenström, Ring of quotients , Springer Verlag, New York, 1975.
[15] C. Vinsonhaler and W. Wickless, The flat dimension of a completely decomposable abelian group as an \(E\)-module , to appear in Proc. Int’l. Conf. on Abelian Groups, Curacao (1991).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.