Perfect Rings for Which the Converse of Schur’s Lemma Holds
نویسندگان
چکیده
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.
منابع مشابه
Extensions of Simple Modules and the Converse of Schur’s Lemma
The converse of Schur’s lemma (or CSL) condition on a module category has been the subject of considerable study in recent years. In this note we extend that work by developing basic properties of module categories in which the CSL condition governs modules of finite length.
متن کاملOn generalizations of semiperfect and perfect rings
We call a ring $R$ right generalized semiperfect if every simple right $R$-module is an epimorphic image of a flat right $R$-module with small kernel, that is, every simple right $R$-module has a flat $B$-cover. We give some properties of such rings along with examples. We introduce flat strong covers as flat covers which are also flat $B$-covers and give characterizations of $A$-perfe...
متن کاملA 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.
متن کاملMinimal Faces and Schur’s Lemma for Embeddings into Ru
In the context of N. Brown’s Hom(N,RU ), we establish that given π : N → RU , the dimension of the minimal face containing [π] is one less than the dimension of the center of the relative commutant of π. We also show the “convex independence”of extreme points in the sense that the convex hull of n extreme points is an n-vertex simplex. Along the way, we establish a version of Schur’s Lemma for ...
متن کاملSelected topics in Mathematical Physics: Quantum Information Theory Talk 5: Averages over the unitary group
In this talk, it is shown how to calculate averages over the Unitary group by applying mathematical tools presented in the previous talk. Particulary, Schur’s Lemma is utilized to two useful examples. Eventually, the results are used to show that a random pure state on a finite-dimensional bipartite Hilbert space is close to maximally entangled. 1 Representations of the Unitary group and Schur’...
متن کامل