Ring Theory Seminar | Quasi-Baer Module Hulls and Applications, April 21

The Ohio University-Ohio State University Ring Theory Seminar series presents “Quasi-Baer Module Hulls and Applications” on Friday, April 21, from 4:45 to 5:45 p.m. in in Cockins Hall 240 in Columbus.

Abstract: For a module $N$, the quasi-Baer hull $\text{\bf qB}(N)$ (resp., the Rickart hull $\text{\bf R}(N)$) of $N$ is the smallest quasi-Baer (resp., Rickart) extension of $N$ if it exists, in a fixed injective hull $E(N)$. In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring $R$ is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective $R$-module has a quasi-Baer hull. Let $R$ be a Dedekind domain with $F$ the field of fractions. Assume that $M_R$ is an $R$-module with $\text{Ann}_R(M)\neq 0$ and $\{K_i\mid i\in\Lambda\}$ is a set of $R$-submodules of $F_R$. Then it is shown that $M_R\oplus (\oplus_{i\in\Lambda}K_i)_R$ has a quasi-Baer module hull if and only if $M_R$ is semisimple. Also the quasi-Baer hull of $M_R\oplus(\oplus_{i\in\Lambda}K_i)_R$ is explicitly described. An example such that $M_R\oplus(\oplus_{i\in\Lambda}K_i)_R$ has no Rickart module hull is provided. As a consequence, for a module $N$ over a Dedekind domain with $N/t(N)$ is projective and $\text{Ann}_R(t(N))\neq 0$, where $t(N)$ is the torsion submodule of $N$, we show that the quasi-Baer hull $\text{\bf qB}(N)$ of $N$ exists if and only if $t(N)$ is semisimple.We also prove the existence of the Rickart hull $\text{\bf R}(N)$ of such $N$. Furthermore, we provide explicit constructions of $\text{\bf qB}(N)$ and $\text{\bf R}(N)$ and show that these two hulls are precisely the same. As applications, it is shown that if $N$ is a finitely generated module over a Dedekind domain, then $N$ is quasi-Baer if and only if $N$ is Baer if and only if $N$ is semisimple or torsion-free. Moreover, if $N$ is a module over a Dedekind domain, which is a direct sum of finitely generated modules, it is shown that $N$ is quasi-Baer if and only if $N$ is Rickart if and only if $N$ is semisimple or torsion-free. The differences between the notion of a quasi-Baer hull and a Baer hull, and that of a quasi-Baer hull and a Rickart hull are exhibited. Also the disparity between Rickart hulls and Baer hulls is shown. Various explicit examples illustrating our results are provided.

(This is a joint work with Gangyong Lee, S. Tariq Rizvi, and Cosmin S. Roman.)