The Algebra Seminar series presents Nguyen Khanh Tung of the University of Padova (Italy) and Ohio University Center of Ring Theory and its Applications.
He will speak on “Rings over Which Every Non-Zero Cyclic Right Module is Poor” on Thursday, Nov. 20, from 4 to 5 p.m. in Morton 215.
Abstract: A module $M$ is said to be poor if it is injective relative to only semisimple modules. A ring $R$ has no right middle class if every right $R$-module is injective or poor. In this talk, we review some properties of poor modules and the characterizations for rings with semisimple poor modules and rings with no middle class. Next, we say that a ring $R$ satisfies $(P)$ if every non-zero cyclic right $R$-module is poor. A non-semisimple ring $R$ with $(P)$ is proved to be an indecomposable ring such that $Z(R_R)$ is essential in $R_R$, every noetherian right$R$-module is artinian, and every ideal of $R$ is either below the prime radical $N$ or above the Jacobson radical $J(R)$. Moreover, we show that a right noetherian ring $R$ with $(P)$ is isomorphic to a matrix ring over a non-uniserial local right artinian ring. In particular, if a commutative noetherian ring $R$ satisfies $(P),$ then $R$ is isomorphic to a direct product of fields ( This is a joint work with Sergio Lopez Permouth and Noyan Er).
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