The Ohio University-Ohio State University ring theory seminar series presents **Dr. Sultan Eylem Toksoy** discussing “Modules with minimal copure-injectivity domain” on Friday, Aug. 31, from 4:45 to 5:45 p.m. in Cockins Hall 240, OSU-Columbus.

Toksoy is Associate Professor of Algebra at Hacettepe University in Ankara, Turkey.

**Abstract**: $R$ will denote an associative ring with identity element and modules will be unital right $R$-modules unless otherwise stated. A right $R$-module $M$ is said to be \emph{cofinitely generated} if $E(M)=E(S_{1})\oplus E(S_{2})\oplus\ldots\oplus E(S_{n})$, where $S_{1}, S_{2},\ldots, S_{n}$ are simple right $R$-modules and $E(X)$ is the injective hull of a right $R$-module $X$ (see \cite{Vamos} and \cite{Jans}). A right $R$-module $M$ is called a \emph{cofree module} if $M$ is isomorphic to $\prod\{E(S_{\alpha})\mid S_{\alpha}$ is a simple right $R$-module, $\alpha\in I\}$ where $I$ is some index set (see \cite{related}). A right $R$-module $M$ is said to be \emph{cofinitely related} if there is an exact sequence $\ShortExactSequence{M}{}{N}{}{K}$ of $R$-modules with $N$ cofinitely generated, cofree and $K$ cofinitely generated (see \cite{related}). In \cite{copure}, the notion of \emph{copurity} has introduced as dual to the notion of purity using cofinitely related modules. A submodule $L$ of a right $R$-module $M$ is said to be \emph{copure} if for every cofinitely related right $R$-module $K$, every homomorphism from $L$ into $K$ has an extension to $M$. A short exact sequence of right $R$-modules $\ShortExactSequence{A}{}{B}{}{C}$ is called a \emph{copure short exact sequence} if every cofinitely related right $R$-module is injective with respect to this sequence. So a submodule $L$ of a right $R$-module $M$ is said to be \emph{copure} in $M$ if the canonical short exact sequence $\ShortExactSequence{L}{}{M}{}{M/L}$ is copure.

Let $M$ and $N$ be right $R$-modules. $M$ is said to be \emph{$N$-copure-injective} if every homomorphism from a copure submodule of $N$ to $M$ can be extended to a homomorphism from $N$ to $M$. $M$ is \emph{copure-injective} if it is injective relative to every copure short exact sequence of right $R$-modules (see \cite{copureinj}).

We define a right $R$-module $M$ as \emph{copure-split} module if every copure submodule of $M$ is a direct summand and we prove that a ring $R$ is a right CDS ring if and only if every $R$-module is copure-injective if and only if every $R$-module is copure-split. We define a right $R$-module $M$ to be \emph{copure-injectively-poor} (simply

\emph{copi-poor}) if the copure-injectivity domain of $M$ is minimal and we study properties of copi-poor modules. Rings over which every right $R$-module is copi-poor is shown to be right CDS rings. In \cite{Ungor}, it is proved that $R$ is a right PDS ring if and only if every right $R$-module is pi-poor. Since commutative PDS rings are CDS (see \cite{copureinj}), a copi-poor module need not be pi-poor in general and conversely. We prove that over commutative (co-)noetherian rings a module is pi-poor if and only if it is copi-poor. Therefore it is obtained that copi-poor Abelian groups coincide with pi-poor Abelian groups.

\begin{thebibliography}{00}

\bibitem{Ungor}

- Harmanc{\i}, S. R. L\'{o}pez-Permouth and B. \”{U}ng\”{o}r, On the pure-injectivity profile of a ring, {\it Comm. Algebra} {\bf 43}(11)

(2015) 4984–5002.

\bibitem{related}

- A. Hiremath, Cofinitely generated and cofinitely related modules, {\it Acta Math. Acad. Sci. Hungar.} {\bf 39} (1982) 1–9.

\bibitem{copure}

- A. Hiremath (Madurai), Copure Submodules, {\it Acta Math. Hung.} {\bf

44}(1-2) (1984) 3–12.

\bibitem{copureinj}

- A. Hiremath, Copure-injective modules, {\it Indian J. Pure Appl.

Math.} {\bf 20}(3) (1989) 250–259.

\bibitem{Jans}

- P. Jans, On co-noetherian rings, {\it J. London Math. Soc.} {\bf 1}

(1969) 588–590.

\bibitem{Vamos}

- Vamos, On the dual of the notion of “finitely generated”, {\it J.

London Math. Soc.} {\bf 43}(1968) 643–646.

\end{thebibliography}

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