Papers by DAVID SSEVVIIRI
arXiv (Cornell University), Sep 27, 2022
Let k be a field, R be the ring k[x, y] and I be a monomial ideal of R. Using combinatorics (in p... more Let k be a field, R be the ring k[x, y] and I be a monomial ideal of R. Using combinatorics (in particular Young diagrams), we characterize, classify and give a geometric interpretation of the largest reduced submodules, R(M ) of the R-modules, M := R/I which when considered as k-modules are finite dimensional.
On the Compactification of Algebraic Curves
Far East Journal of Mathematical Sciences, Jul 16, 2018
arXiv (Cornell University), May 26, 2022
This is the first in a series of papers highlighting the applications of reduced and coreduced mo... more This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let R be a commutative unital ring and I an ideal of R. We show that I-reduced R-modules and I-coreduced R-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the I-torsion and the I-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.
International Electronic Journal of Algebra, 2016
The formal study of completely prime modules was initiated by N. J. Groenewald and the current au... more The formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper; Completely prime submodules,
International Journal of Algebra, 2013
We characterize non-nilpotent elements of the Z-module D K = Z/(p k 1 1 × • • • × p kn n)Z. The p... more We characterize non-nilpotent elements of the Z-module D K = Z/(p k 1 1 × • • • × p kn n)Z. The projective limit of non-nilpotent elements of the Zmodule D K adjoined with zero is an ideal of the ring n i=1 Z p i , where Z p i is the ring of p i-adic integers. It turns out that the results that were obtained in [5] are just corollaries and Question 2.1 that was posed in [5] is also answered.
arXiv (Cornell University), Dec 9, 2016
We introduce a complete radical formula for modules over non-commutative rings which is the equiv... more We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known results about modules over commutative rings that satisfy the radical formula are retrieved. Examples and properties of modules that satisfy the complete radical formula are given. For instance, it is shown that a module that satisfies the complete radical formula is completely semiprime if and only if it is a subdirect product of completely prime modules. This generalizes a ring theoretical result: a ring is reduced if and only if it is a subdirect product of domains. We settle in affirmative a conjecture by Groenewald and the current author given in [5] that a module over a 2-primal ring is 2-primal. More instances where 2-primal modules behave like modules over commutative rings are given. This is in tandem with the behaviour of 2-primal rings of exhibiting tendencies of commutative rings. We end with some questions about the role of 2-primal rings in algebraic geometry.
arXiv (Cornell University), Jun 22, 2023
This is the second in a series of papers highlighting the applications of reduced and coreduced m... more This is the second in a series of papers highlighting the applications of reduced and coreduced modules. Let R be a commutative unital ring and I be an ideal of R. We give the necessary and sufficient conditions in terms of I-reduced and I-coreduced R-modules for the functors Hom R (R/I, −) and Γ I , the I-torsion functor, on the abelian full subcategories of the category of all R-modules to be radicals. These conditions: 1) subsume and unify many results which were proved on a case-by-case basis, 2) provide a setting for the generalisation of Jans' correspondence of an idempotent ideal of a ring with a torsion-torsionfree class, 3) provide answers to open questions that were posed by Rohrer, and 4) lead to a new radical class of rings.
International Electronic Journal of Algebra
Different and distinct notions of regularity for modules exist in the literature. When these noti... more Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.
Rendiconti del Circolo Matematico di Palermo Series 2
Let R be a commutative ring, M an R-module and ϕ a be the endomorphism of M given by right multip... more Let R be a commutative ring, M an R-module and ϕ a be the endomorphism of M given by right multiplication by a ∈ R. We say that M is weaklymorphic if M/ϕ a (M) ∼ = ker(ϕ a) as R-modules for every a. We study these modules and use them to characterise the rings R/Ann R (M), where Ann R (M) is the right annihilator of M. A kernel-direct or image-direct module M is weakly-morphic if and only if each element of R/Ann R (M) is regular as an endomorphism element of M. If M is a weakly-morphic module over an integral domain R, then M is torsion-free if and only if it is divisible if and only if R/Ann R (M) is a field. A finitely generated Z-module is weakly-morphic if and only if it is finite; and it is morphic if and only if it is weakly-morphic and each of its primary components is of the form (Z p k) n for some non-negative integers n and k.
Rendiconti del Circolo Matematico di Palermo Series 2
Let R be a commutative unital ring, a ∈ R and t a positive integer. a t-reduced R-modules and uni... more Let R be a commutative unital ring, a ∈ R and t a positive integer. a t-reduced R-modules and universally a t-reduced R-modules are defined and their properties given. Known (resp. new) results about reduced R-modules are retrieved (resp. obtained) by taking t = 1 and results about reduced rings are deduced.
arXiv (Cornell University), Dec 11, 2018
A relationship between nilpotency and primeness in a module is investigated. Reduced modules are ... more A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural properties. A general form is given of an example used in literature to distinguish: 1) completely prime modules from prime modules, 2) classical prime modules from classical completely prime modules, and 3) a module which satisfies the complete radical formula from one which is neither 2-primal nor satisfies the radical formula.
arXiv (Cornell University), Dec 9, 2016
We introduce a complete radical formula for modules over non-commutative rings which is the equiv... more We introduce a complete radical formula for modules over non-commutative rings which is the equivalence of a radical formula in the setting of modules defined over commutative rings. This gives a general frame work through which known results about modules over commutative rings that satisfy the radical formula are retrieved. Examples and properties of modules that satisfy the complete radical formula are given. For instance, it is shown that a module that satisfies the complete radical formula is completely semiprime if and only if it is a subdirect product of completely prime modules. This generalizes a ring theoretical result: a ring is reduced if and only if it is a subdirect product of domains. We settle in affirmative a conjecture by Groenewald and the current author given in [5] that a module over a 2-primal ring is 2-primal. More instances where 2-primal modules behave like modules over commutative rings are given. This is in tandem with the behaviour of 2-primal rings of exhibiting tendencies of commutative rings. We end with some questions about the role of 2-primal rings in algebraic geometry.
Journal of Algebra and Its Applications, May 7, 2013
In this paper, the concept of 2-primal modules is introduced. We show that the implications betwe... more In this paper, the concept of 2-primal modules is introduced. We show that the implications between rings which are reduced, IFP, symmetric and 2primal are preserved when the notions are extended to modules. Like for rings, for 2-primal modules, prime submodules coincide with completely prime submodules. We prove that if M is a quasi-projective and finitely generated right R-module which is a self-generator, then M is 2-primal if and only if S =End R (M) is 2primal. Some properties of 2-primal modules are also investigated.
The formal study of completely prime modules was initiated by N. J. Groenewald and the current au... more The formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper; Completely prime submodules, Int. Elect. J. Algebra, 13, (2013), 1–14. In this paper, the study of completely prime modules is continued. Firstly, the advantage completely prime modules have over prime modules is highlited and different situations that lead to completely prime modules given. Later, emphasis is put on fully completely prime modules, (i.e., modules whose all submodules are completely prime). For a fully completely prime left R-module M , if a, b ∈ R and m ∈ M , then abm = bam, am = am for all positive integers k, and either am = abm or bm = abm. In the last section, two different torsion theories induced by the completely prime radical are given.
arXiv: Commutative Algebra, 2017
We show that existence of nonzero nilpotent elements in the $\Z$-module $\Z/(p_1^{k_1}\times \cdo... 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.
On the Compactification of Algebraic Curves
Far East Journal of Mathematical Sciences (FJMS), 2018
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2020
Let R be a commutative unital ring and a ∈ R. We introduce and study properties of a functor aΓ a... more Let R be a commutative unital ring and a ∈ R. We introduce and study properties of a functor aΓ a (−), called the locally nilradical on the category of R-modules. aΓ a (−) is a generalisation of both the torsion functor (also called section functor) and Baer's lower nilradical for modules. Several local-global properties of the functor aΓ a (−) are established. As an application, results about reduced R-modules are obtained and hitherto unknown ring theoretic radicals as well as structural theorems are deduced.
International Electronic Journal of Algebra, 2014
In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we de... more 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 R M with RM = 0 forms a special class of modules.
International Electronic Journal of Algebra, 2015
The coincidence of the set of all nilpotent elements of a ring with its prime radical has a modul... more The coincidence of the set of all nilpotent elements of a ring with its prime radical has a module analogue which occurs when the zero submodule satisfies the radical formula. A ring R is 2-primal if the set of all nilpotent elements of R coincides with its prime radical. This fact motivates our study in this paper, namely; to compare 2-primal submodules and submodules that satisfy the radical formula. A demonstration of the importance of 2-primal modules in bridging the gap between modules over commutative rings and modules over noncommutative rings is done and new examples of rings and modules that satisfy the radical formula are also given.
International Journal of Algebra, 2013
We characterize non-nilpotent elements of the Z-module D K = Z/(p k 1 1 × • • • × p kn n)Z. The p... more We characterize non-nilpotent elements of the Z-module D K = Z/(p k 1 1 × • • • × p kn n)Z. The projective limit of non-nilpotent elements of the Zmodule D K adjoined with zero is an ideal of the ring n i=1 Z p i , where Z p i is the ring of p i-adic integers. It turns out that the results that were obtained in [5] are just corollaries and Question 2.1 that was posed in [5] is also answered.
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Papers by DAVID SSEVVIIRI