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Automorphism-invariant modules. (English) Zbl 1333.16006

In this quite interesting paper the authors study automorphism-invariant (auto-inv., in short, in this review) modules. The authors investigate sufficient conditions for an auto-inv. module to be quasi-injective. For convenience, in this review, \(M\) stands for an auto-inv. module and \(E(M)\) for its injective envelope. The authors prove that if under the canonical ring embedding of \(\mathrm{End}(M)/J(\mathrm{End}(M))\) into \(\mathrm{End}(E(M))/J(\mathrm{End}(E(M)))\) every idempotent of \(\mathrm{End}(E(M))/J(\mathrm{End}(E(M)))\) has an idempotent pre-image in \(\mathrm{End}(M)/J(\mathrm{End}(M))\) then \(M\) is quasi-injective. They also prove, for a ring \(R\), that every auto-inv. \(R\)-module of finite Goldie dimension is quasi-injective if and only if every indecomposable, auto-inv. \(R\)-module of finite Goldie dimension is uniform.
Next, the authors study the connection between auto-inv. modules and Boolean rings. They prove that if \(M\) is a square-free, auto-inv. module then \(\mathrm{End}(M)/J(\mathrm{End}(M))\) is Abelian, von Neumann regular ring. The authors conclude the paper with the following result. If \(M\) is an auto-inv. square-free module of finite Goldie dimension then \(M=N\oplus P\) where \(N,P\) are orthogonal submodules of \(M\) with the properties that \(\mathrm{End}(N)\) has no factor isomorphic to the 2-element field \(\mathbb F_2\) and \(\mathrm{End}(P)/J(\mathrm{End}(P))\) is isomorphic to a Boolean ring \((\mathbb F_2)^n\) for some \(n\in\mathbb N\). The authors also prove that if \(M\) is an auto-inv. module with the additional condition that either \(M\) is of finite Goldie dimension or \(M\) is square-free then \(M\) is co-Hopfian, that is, every injective endomorphism of \(M\) is an automorphism of \(M\).
To conclude I add following two comments. (c) of Lemma 3.1 follows just by the choice of the ideal \(I\). Towards the end of the proof of Proposition 3.10, \(e_1-e_2\) need not be an idempotent even though \(e_1,e_2\) are orthogonal. So \(e_1-e_2\) should be replaced by \(e_1+e_2\) and hence we get \(e_1=-e_2\) and rest of the proof remains the same.

MSC:

16D50 Injective modules, self-injective associative rings
16W20 Automorphisms and endomorphisms
16S50 Endomorphism rings; matrix rings
16U80 Generalizations of commutativity (associative rings and algebras)
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