Prime Elements in Partially Ordered Groupoids Applied to Modules and Hopf Algebra Actions

Abstract

Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules L(M ) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which L(M ) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz' "weakly compressible" modules. In particular we are interested in representing weakly compressible modules as a subdirect product of "prime" modules in a suitable sense. It turns out that any weakly compressible module is a subdirect product of prime modules (in the sense of Kaplansky). Moreover if M is a self-projective module, then M is weakly compressible if and only if it is a subdirect product of prime modules (in the sense of Bican et al.). An application to Hopf actions is given.

On Subdirect Product of Prime Modules

“…In the literature, there are several module generalizations of a semiprime (prime) ring, see [15,Sections 13 and 14] for an excellent reference on the subject. These generalizations introduce various concepts of semiprime (prime) modules and many important theories on semiprime (prime) rings are generalized to modules by them, see; [3], [7], [8], [10] and [17]. The natural question "when are semiprime modules subdirect product of primes?"…”

“…Then we illustrate the main results about the above question. Following [10], a module M R is called ⋆-prime if M ∈ Cog(N ) for any 0 = N ≤ M R . These modules were originally studied in [4].…”

“…A widely used generalization of semiprime rings is weakly compressible modules M R that was defined in [2] by Hom R (M, N )N = 0 for all 0 = N ≤ M R . They are semiprime in the sense of [10] (i.e., M ∈ Cog(N ) 278 N. DEHGHANI AND M. R. VEDADI for every N ≤ ess M R ). As we will see in Proposition 2.1, semiprime R-modules have the property " ann R (N ) is a semiprime ideal of R for all 0 = N ≤ M R ".…”

“…In [5, Theorems 2.6 and 3.3], the conditions S ⊆ Ω P⋆ and W = Ω P⋆ are investigated for certain duo rings. In [10,Corollary 5.4], it is shown that W ⊆ Ω P . The aim of this paper is study the equality "W = Ω P " for commutative rings (or more generally duo rings) as stated in the abstract of the paper.…”

Semiprime and Weakly Compressible Modules

“…These semiprime modules are precisely weakly compressible modules in the sense of [1]; see for example Theorem 2.5 below. Following [6], a module M R is called weakly compressible if Hom R (M, N )N ̸ = 0 for all nonzero N ≤ M R . We also call M R semiprime if every essential submodule of M R cogenerates M R .…”