Trace ideals and centers of endomorphism rings of modules over commutative rings. (English) Zbl 1367.16026
Let \(R\) be a commutative ring and \(M\) be a finitely generated \(R\)-module. Let \(M^*={\mathrm{Hom}_R}(M,R)\) and \(\tau_M(R)= \sum_{\alpha\in M^*} \alpha(M)\) be the trace ideal of \(M\). Also assume that the center of a non-commutative ring \(S\) is denoted by \({\mathrm{Z}}(S)\). We say that \(M\) is reflexive when the natural evaluation map \(M\to M^{**}\) is an isomorphism. In this paper, under certain conditions, the author constructs \(R\)-algebra monomorphisms from \({\mathrm{End}_R}(\tau_M(R))\) to \({\mathrm{Z}}({\mathrm{End}_R}(M))\) or \({\mathrm{Z}}({\mathrm{End}_R}(M^*))\) and conversely from the center of each of these endomorphism rings to \({\mathrm{End}_R}(\tau_M(R))\). In particular, it is shown that if \(R\) is Noetherian and \(M\) is reflexive and faithful then \({\mathrm{End}_R}(\tau_M(R))\cong {\mathrm{Z}}({\mathrm{End}_R}(M))\) and if \(R\) is Noetherian and \(\tau_M(R)\) contains a non-zero-divisor, then \({\mathrm{End}_R}(\tau_M(R))\cong {\mathrm{End}_R}(\tau_{M^*}(R))\cong {\mathrm{Z}}({\mathrm{End}_R}(M^*))\), as \(R\)-algebras.
Then these results is applied to study rigid modules over one-dimensional Gorenstein rings and balanced modules. The module \(M\) is called rigid when \({\mathrm{Ext}_R}^1(M,M)=0\) and it is called balanced if \(R={\mathrm{Z}}({\mathrm{End}_R}(M))\). Among some other results, the author shows that if \(R\) is a Noetherian local ring of depth \(\leq 1\) and \(M\) is reflexive, then \({\mathrm{End}_R}(M)\) has a free summand if and only if \(M\) has a free summand if and only if \(M\) is balanced. Also under these assumptions \(M\) is free if and only if \({\mathrm{End}_R}(M)\) is so.
It is also proved that if \(R\) is a one-dimensional Gorenstein local ring and \(M\) is torsion-free, faithful and rigid and if \({\mathrm{Z}}({\mathrm{End}_R}(M))\) is Gorenstein, then \(M\) has a free summand. Moreover, the author shows how this result is related to a conjecture of C. Huneke and R. Wiegand in [Math. Ann. 299, No. 3, 449–476 (1994; Zbl 0803.13008)]. This conjecture states that if \(M\) is a finitely generated torsion-free module over a Gorenstein local domain of dimension 1, and if \(M\) is not free, then \(M\otimes_R M^*\) is not torsion-free.
Finally, I should mention that this paper contains most of the needed preliminary results and their proofs and hence, to a great extent, it is self-content.
Then these results is applied to study rigid modules over one-dimensional Gorenstein rings and balanced modules. The module \(M\) is called rigid when \({\mathrm{Ext}_R}^1(M,M)=0\) and it is called balanced if \(R={\mathrm{Z}}({\mathrm{End}_R}(M))\). Among some other results, the author shows that if \(R\) is a Noetherian local ring of depth \(\leq 1\) and \(M\) is reflexive, then \({\mathrm{End}_R}(M)\) has a free summand if and only if \(M\) has a free summand if and only if \(M\) is balanced. Also under these assumptions \(M\) is free if and only if \({\mathrm{End}_R}(M)\) is so.
It is also proved that if \(R\) is a one-dimensional Gorenstein local ring and \(M\) is torsion-free, faithful and rigid and if \({\mathrm{Z}}({\mathrm{End}_R}(M))\) is Gorenstein, then \(M\) has a free summand. Moreover, the author shows how this result is related to a conjecture of C. Huneke and R. Wiegand in [Math. Ann. 299, No. 3, 449–476 (1994; Zbl 0803.13008)]. This conjecture states that if \(M\) is a finitely generated torsion-free module over a Gorenstein local domain of dimension 1, and if \(M\) is not free, then \(M\otimes_R M^*\) is not torsion-free.
Finally, I should mention that this paper contains most of the needed preliminary results and their proofs and hence, to a great extent, it is self-content.
Reviewer: A. Nikseresht (Zanjan)
MSC:
| 16S50 | Endomorphism rings; matrix rings |
| 13C13 | Other special types of modules and ideals in commutative rings |
Citations:
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