On Loewy length of rings

Abstract

Associated with each ring R over which every nonzero right module has a minimal submodule is an ordinal number called its right (lower) Loewy length. The concern here is with the various possible left and right Loewy lengths of such rings with zero radical and with the possible right-left symmetry of this minimal submodule condition. In particular, if R has finite right Loewy length n then R has left Loewy length g 2n — 1. All rings are associative rings with identity. Denoting the socle of a module M by Soc (M), the (lower) Loewy series for M is defined transfinitely by: So = 0, Sa+JSa = Soc (M/Sa) and, if a is a limit ordinal, Sa = \Jβ<aSβ. (See Bass [1, p. 470].) If M = Sa for some ordinal number a then M is called a (lower) Loewy module. The Loewy length of such a module is L(M) = 7, the least ordinal 7 with M — Sr. We call a ring R a right (resp., left) Loewy ring in case the regular representation RR (resp., RR) is a Loewy module. (Nastasescu and Popescu [6] use the term semi-artinian to denote such a ring.) A ring R is easily seen to be a right Loewy ring if and only if each of its right modules has a nontrivial (hence, essential) socle. Over such a ring each right module M is a Loewy module of length L(M) ^ L(RR) and, since RR is finitely generated, L(RR) cannot be a limit ordinal.