The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.

Key Words:

• Group algebra
• Perfect rings
• Primary decomposable
• Schur's Lemma
• Sylow-p-subgroup

1991 Mathematics Subject Classification:

• Primary 16S50
• Secondary 16P20, 16P40, 16S34

ACKNOWLEDGMENT

The authors wish to thank the referee for his helpful suggestions.

This work was partially supported by El Minesterio de Ciencia y tecnologia proyecto BFM 2001-2335. Spain.

Notes

^#Communicated by by T.Lenagan.