A complete radical formula and 2-primal modules

Abstract

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.

Nilpotent elements control the structure of a module

“…5. It is an example of a module that satisfies the complete radical formula but it is neither 2-primal nor satisfies the radical formula; see a particular case used in [41].…”

“…It is an example of a prime nil module. It is a general form of the example which was used in [40] to distinguish a prime module from a completely prime module, used in [13] to distinguish a classical prime module from a classical completely prime module and an example used in [41] of a module which satisfies the complete radical formula but it is neither 2-primal nor satisfies the radical formula. It is an example which shows another advantage (in addition to those already known and given in [40]) that completely prime modules have over prime modules.…”