- Department of Mathematics,
College of Natural Sciences
Makerere University
website: www.math.mak.ac.ug
P.O BOX 7062
Kampala Uganda
Ssevviiri David
Makerere University, Mathematics, Faculty Member
We show that existence of nonzero nilpotent elements in the $\Z$-module $\Z/(p_1^{k_1}\times \cdots \times p_n^{k_n})\Z$ inhibits the module from possessing good structural properties. In particular, it stops it from being semisimple and... more
We show that existence of nonzero nilpotent elements in the $\Z$-module $\Z/(p_1^{k_1}\times \cdots \times p_n^{k_n})\Z$ inhibits the module from possessing good structural properties. In particular, it stops it from being semisimple and from admitting certain good homological properties.
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A notion of 2-primal rings is generalized to modules by defining 2-primal modules. We show that the implications between rings which are reduced, have insertion-of-factor-property (IFP), reversible, semi-symmetric and 2-primal are... more
A notion of 2-primal rings is generalized to modules by defining 2-primal modules. We show that the implications between rings which are reduced, have insertion-of-factor-property (IFP), reversible, semi-symmetric and 2-primal are preserved when the notions are extended to modules. Like for rings, 2-primal modules bridge the gap between modules over commutative rings and modules over non-commutative rings; for instance, for 2-primal modules, prime submodules coincide with completely prime submodules. Completely prime submodules and reduced modules are both characterized. A generalization of 2-primal modules is done where 2-primal and NI modules are a special case.
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All modules considered are left modules which are not necessarily unital. The rings are associative. We write A ⊲ R to mean A is an ideal of R and N ≤ M to mean N is a submodule of M. A nonzero element m of an R-module M is nilpotent [2]... more
All modules considered are left modules which are not necessarily unital. The rings are associative. We write A ⊲ R to mean A is an ideal of R and N ≤ M to mean N is a submodule of M. A nonzero element m of an R-module M is nilpotent [2] of degree k if there exists a ...
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ABSTRACT We define nilpotent and strongly nilpotent elements of a module M and show that the set s (M) of all strongly nilpotent elements of M over a commutative unital ring R coincides with the classical prime radical βcl (M) the... more
ABSTRACT We define nilpotent and strongly nilpotent elements of a module M and show that the set s (M) of all strongly nilpotent elements of M over a commutative unital ring R coincides with the classical prime radical βcl (M) the intersection of all classical prime submodules of M.
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Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{... more
Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$ . It is shown that $\mathcal{S}(N)$ is equal to the intersection of all s-prime submodules of M containing N. We define $\mathcal{N}({}_{R}M) = \mathcal{S}(0)$ . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then $\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ where ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ denotes the prime radical of M. We also show that if R is a left Artinian ring, then ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)M$ where ${\mathop{\mathrm{Rad}}\nolimits}\, (M)$ denotes the Jacobson radical of M and ${\mathop{\mathrm{Jac}}\nolimits}\, (R)$ the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.