Extensions of Simple Modules and the Converse of Schur’s Lemma

Perfect Rings for Which the Converse of Schur’s Lemma Holds

If M is a simple module over a ring R then, by the Schur’s lemma, the endomorphism ring of M is a division ring. However, the converse of this result does not hold in general, even when R is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.

A basic note on group representations and Schur’s lemma

Here we look at some basic results from group representation theory. Moreover, we discuss Schur’s Lemma in the context of R[G]-modules and provide some specialized results in that case.

Extensions of Baer and quasi-Baer modules

Inert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules

Introduction Suppose that  is a commutative ring with identity,  is a unitary -module and  is a multiplicatively closed subset of .  Factorization theory in commutative rings, which has a long history, still gets the attention of many researchers. Although at first, the focus of this theory was factorization properties of elements in integral domains, in the late nineties the theory was gener...

A simple proof of Zariski's Lemma

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.