ON THE LEVITZKI RADICAL OF MODULES

Year 2014, Volume: 15 Issue: 15, 77 - 89, 01.06.2014

Nico J. Groenewald David Ssevviiri

https://doi.org/10.24330/ieja.266239

Cited By: 1

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.

Keywords

l-prime submodule, semi l-prime, s-prime submodule, upper nil radical, Levitzki radical

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• Department of Mathematics and Applied Mathematics Nelson Mandela Metropolitan University Port Elizabeth South Africa e-mail: [e-mail] David Ssevviiri Department of Mathematics and Applied Mathematics Nelson Mandela Metropolitan University Port Elizabeth South Africa e-mail: [e-mail]