Primeness described in the language of torsion preradicals

Abstract

The objects of study in this paper are lattice ordered monoids. These are structures L; ⊕, 0 L ; ≤ where L; ⊕ , 0 L is a monoid, L; ≤ is a lattice and the binary operation ⊕ distributes over finite meets. If R is an arbitrary ring with identity then the set Id R of all ideals of R and the set torsp R of all torsion preradicals on the category of right R -modules, are examples of lattice ordered monoids. Primeness and semiprimeness are ring theoretic notions which are characterizable as first order sentences in the language of Id R . The main results show that the notions right strong primeness and right strong semiprimeness may be characterized in a variety of ways as first order sentences in the language of torsp R .

Duprime and dusemiprime modules

“…Interpreting [11,Theorem 14] in the case where the lattice ordered monoid is chosen to be IL M , we obtain:…”

“…In particular, taking P to be the zero ideal, R will be a prime ring if the hereditary pretorsion class consisting of all left R-modules, namely R-Mod, is duprime in IL R . It is shown in [11,Theorem 26 and Remark 27] that the rings R for which R-Mod is duprime are precisely the left strongly prime rings of Handelman and Lawrence [7]. It is shown similarly [11,Theorem 32 and Remark 33] that R-Mod is dusemiprime if and only if R is left strongly semiprime in the sense of Handelman [6].…”

“…Results in this paper have a mixed flavour; they make use of standard module theoretic techniques, but are also reliant on the body of theory on lattice ordered monoids developed in [11].…”

“…Notice that the operation ':' defined here is opposite to the multiplication operation introduced in[12],[13] and[11]. Consequently, properties which are prefixed with 'left' in the aforementioned papers, become 'right' in this paper.…”

On the Splitting Property of Rings with Restricted Class of Injectivity Domains

“…For any given ring R, we denote the class of all hereditary pretorsion classes of right R-modules by hptors-R. Note that hptors-R has a natural lattice structure, with the partial order given by inclusion, which is coatomic ( [22,Theorem 2]). As shown in [15, Theorem 2.9], the profile of the ring R, denoted by iP(R), is equal to the interval [SSMod-R, Mod-R] = {T ∈ hptors-R : SSMod-R ⊆ T }, a coatomic sublattice of hptors-R.…”