The locally nilradical for modules over commutative rings
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The locally nilradical for modules over commutative rings Annet Kyomuhangi1 · David Ssevviiri1 Received: 29 October 2019 / Accepted: 17 February 2020 © The Managing Editors 2020
Abstract Let R be a commutative unital ring and a ∈ R. We introduce and study properties of a functor aa (−), called the locally nilradical on the category of R-modules. aa (−) 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 aa (−) 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. Keywords Locally nilradical · Baer’s lower nilradical · Torsion functor · Reduced modules · Reduced rings · Local cohomology Mathematics Subject Classification 16S90 · 16N80 · 13D45
1 Introduction Radicals are a good tool to study the structure of rings and modules over rings. There are several radicals in the literature about rings and modules which include among others; Baer’s lower nilradical (also called the prime radical), Köthe’s upper nilradical, Andrunakievich’s generalised nilradical (also called the completely prime radical), Jacobson radical and Brown–McCoy radical. In this article, we introduce and study a radical called the locally nilradical for modules over commutative rings. Radical theory also exists for abelian categories, and it is what is termed as torsion theory. Throughout this paper, all rings R are commutative and unital. √ The category of all (0 : M) denotes the left R-modules is denoted by√R-Mod. If M ∈ R-Mod, then radical ideal of (0 : M), i.e., (0 : M) = r ∈ R | r k ∈ (0 : M) for some k ∈ Z+ .
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David Ssevviiri [email protected] Annet Kyomuhangi [email protected]
1
Department of Mathematics, Makerere University, P.O BOX 7062, Kampala, Uganda
123
Beitr Algebra Geom
Let M be an R-module and a an ideal of R. The a-torsion (also called the section) functor is defined by: a : R-Mod → R-Mod M → a (M), where a (M) is the submodule of M given by a (M) := m ∈ M | ak m = 0 for some k ∈ Z+ . On modules defined over Noetherian rings, this functor is left exact and a radical, see Rohrer (2018). Its right derived functor Hai (−) is what is called the local cohomology functor with respect to a. For more information about local cohomology, see Brodmann and Sharp (2013). If R is a ring, a ∈ R and a an ideal of R generated by a, then it is easy to see that a (M) = a (M), where a (M) := m ∈ M | a k m = 0 for some k ∈ Z+ . By generalising the torsion functor, we define a new functor: aa : R-Mod → R-Mod M → aa (M), called the locally nilradical which associates toevery R-module M a submodule aa (M)for every a ∈ R, where aa (M) := am | a k m = 0, m ∈ M, for some k ∈ Z+ , i.e., left multiplication by a of the submodule a (M). aa (M) is contained in the envelope E M (0) of M which has been considered in the literature as a module analogue of the set of nilpotent elements of
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