Ring Theory Seminar - Sultan Eylem Toksoy

Title: Modules with minimal copure-injectivity domain

Speaker: Sultan Eylem Toksoy (Hacettepe University, 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 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 [6] and [5]). A right $R$-module $M$ is called a 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 cofinitely related if there is an exact sequence $M \rightarrow N \rightarrow K$ of $R$-modules with $N$ cofinitely generated, cofree and $K$ cofinitely generated (see [2]). In [3], the notion of 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 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 $A \rightarrow B \rightarrow C$ is called a 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 copure in $M$ if the canonical short exact sequence $ L \rightarrow M \rightarrow M/L$ is copure.

Let $M$ and $N$ be right $R$-modules. $M$ is said to be $N$-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 copure-injective if it is injective relative to every copure short exact sequence of right $R$-modules (see [4]). We define a right $R$-module $M$ as 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 copure-injectively-poor (simply 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 [1], 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 [4]), 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.

Bibliography

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