ON THE LEVITZKI RADICAL OF MODULES

Abstract

In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we define and characterize l-prime submodules. Let N be a submodule of an R-module M. If l.√N := {m ∈ M : every l- system of M containingm meets N}, we show that l.√N coincides with the intersection L(N) of all l-prime submodules of M containing N. We define the Levitzki radical of an R-module M as L(M) = l.√0. Let β(M), U(M) and Rad(M) be the prime radical, upper nil radical and Jacobson radical of M respectively. In general β(M) ⊆ L(M) ⊆ U(M) ⊆ Rad(M). If R is commutative, β(M) = L(M) = U(M) and if R is left Artinian, β(M) = L(M) = U(M) = Rad(M). Lastly, we show that the class of all l-prime modules RM with RM 6= 0 forms a special class of modules.