The Ohio University and Ohio States University ring theory seminar presents Nguyen Khanh Tung on “Automorphism-invariant modules” on Friday, Oct. 10, at 4:45 p.m. at the Department of Mathematics at Ohio State University in Columbus.
Nguyen is from the University of Padova (Italy) and is a member of the Ohio University Center of Ring Theory and its Applications.
Abstract: A module $M$ is called automorphism-invariant if it is invariant under automorphisms of its injective envelope. In this talk, we study the endomorphism rings of automorphism-invariant modules and their injective envelopes. We investigate some cases where automorphism-invariant modules are quasi-injective and a connection between automorphism-invariant modules and boolean rings.
We show that if $M$ is an automorphism-invariant module and $E(M)$ is an injective envelope of $M$, then there is a local morphism $\varphi: End(M) \to End(E(M))$ with kernel $J(End(M))$, so that $\varphi$ induces an embedding $\overline{\varphi}$. Hence $End(M)/J(End(M)$ is a rationally closed subring of $End(E(M))/J(End(E(M)))$. Also we prove that an automorphism invariant module $M$ is quasi-injective if and only if for every idempotent $f$ of the ring $End(E(M))/J(End(E(M))$ there exists an idempotent $e$ of $End(M)/J(End(M))$ such that $\overline{\varphi}(e)=f$. Additionally, we consider the cases of automorphism-invariant modules of finite Goldie dimension or indecomposable. In the former, $End(M)$ is a semiperfect ring. In the latter, $End(M)$ is a local ring. Besides, if $M$ is an automorphism-invariant square-free module of finite Goldie dimension, $M$ decomposes as a direct sum $M=N \oplus P$ where $N$ is orthogonal to $P$, $End(N)$ has no factor isomorphic to $mathbb{F}_2$ and $End(P)/J(End(P))$ is isomorphic to a boolean ring $\mathbb{F}_2^n$ for some $n$. (This is a joint work with Adel Alahmadi and Alberto Facchini).
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