ON THE LEVITZKI RADICAL OF MODULES

Authors : Nico J GROENEWALD, David SSEVVİİRİ

Pages : 77-89

Doi:10.24330/ieja.266239

View : 30 | Download : 9

Publication Date : 2014-06-01

Article Type : Research Paper

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 Linsert ignore into journalissuearticles values(N); of all l-prime submodules of M containing N. We define the Levitzki radical of an R-module M as Linsert ignore into journalissuearticles values(M); = l.√0. Let βinsert ignore into journalissuearticles values(M);, Uinsert ignore into journalissuearticles values(M); and Radinsert ignore into journalissuearticles values(M); be the prime radical, upper nil radical and Jacobson radical of M respectively. In general βinsert ignore into journalissuearticles values(M); ⊆ Linsert ignore into journalissuearticles values(M); ⊆ Uinsert ignore into journalissuearticles values(M); ⊆ Radinsert ignore into journalissuearticles values(M);. If R is commutative, βinsert ignore into journalissuearticles values(M); = Linsert ignore into journalissuearticles values(M); = Uinsert ignore into journalissuearticles values(M); and if R is left Artinian, βinsert ignore into journalissuearticles values(M); = Linsert ignore into journalissuearticles values(M); = Uinsert ignore into journalissuearticles values(M); = Radinsert ignore into journalissuearticles values(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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