Title: A Family of Examples of Generalized Perfect Rings
Speaker: Pınar Aydoğdu (Hacettepe University, Ankara, Turkey)
Abstract: Inspired by the fundamental work of Bass [2] on perfect rings and projective covers, A.~Amini, B.~Amini, M.~Ershad and H.~Sharif proposed in [1] to study a class of rings that they named generalized perfect rings.
Let $R$ be a ring, and let $F$ and $M$ be right $R$-modules such that $F_R$ is flat. Following [1], a module epimorphism $f\colon F\rightarrow M$ is said to be a $G$-flat cover of $M$ if $\mathrm{Ker}\, (f)$ is a small submodule of $F$. Still following [1], a ring $R$ is called right generalized perfect (right $G$-perfect, for short) if every right $R$-module has a flat cover. A ring $R$ is called $G$-perfect if it is both left and right $G$-perfect. It is clear from the definition that right perfect rings are right $G$-perfect rings, and also that von Neumann regular rings are $G$-perfect rings.
Looking for a characterization of $G$-perfect rings, it was showed in [1] that if $R$ is right $G$-perfect, then the Jacobson radical $J(R)$ is right $T$-nilpotent and, hence, idempotents lift modulo $J(R)$. Moreover, it was also proved that if $R$ is right duo (i.e. all right ideals are two-sided ideals) and right $G$-perfect, then $R/J(R)$ is von Neumann regular. It was claimed that it was reasonable to conjecture that a right $G$-perfect ring is von Neumann regular modulo the Jacobson radical. In this work, we answer this conjecture in the negative by constructing semiprimitive $G$-perfect rings that are not von Neumann regular. (This is a joint work with Dolors Herbera.)
- A. Amini, B. Amini, M. Ershad and H. Sharif. On generalized perfect rings, Comm. Algebra 35 (2007), 953--963.
- H. Bass. Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466--488.
