A module M is called product closed if every hereditary pretorsion class in σ [M] is closed under products in σ [M]. Every module M which is locally of finite length (every finitely generated submodule of M has finite length) is product closed and every product closed module M is semilocal (M/J (M) is semisimple). Let M ∈ R-Mod be product closed and projective in σ [M]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invariant submodules; (3) M has finite length if M is finitely generated and every hereditary pretorsion class in σ [M] is M-dominated. If the ring R is commutative it is proven that M is product closed if and only if M is locally of finite length. It was shown by Beachy and Blair [2, Proposition 1.4 and Corollary 3.3] that the following three conditions on a ring R with identity are equivalent: (1) every hereditary pretorsion class in R-Mod is closed under arbitrary (and not just finite) direct products, or equivalently, every left topologizing filter on R is closed under arbitrary (and not just finite) intersections; (2) every left R-module M is finitely annihilated, meaning (0 : M) = (0 : X ) for some finite subset X of M; (3) R is left artinian. In this paper we shall attempt to describe those modules M with the property that every hereditary pretorsion class in the Grothendieck category σ [M] is closed under products in σ [M]. One of our main results says that if M is a finitely generated product closed module such that M is projective in σ [M] and every hereditary pretorsion class in σ [M] is M-dominated (meaning, every hereditary pretorsion class in σ [M] is subgenerated by an M-generated module), then M has finite length. This result extends Beachy and Blair's characterization of left artinian rings. Their proof is based on two results due to Beachy [1, Propositions 1 and 5], but the techniques used by Beachy are not easily generalized in a manner useful for our purposes. We have thus had to develop new methods.
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 .
A nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset X of R we have xXy ^ 0 whenever 0 ^ x, y € R. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n = 1. This paper is devoted primarily to an investigation of uniform bounds of primeness in matrix rings over fields. It is shown that the existence of certain n -dimensional nonassociative algebras over a field F decides the uniform bound of the n x n matrix ring over F.1991 Mathematics subject classification (Amer. Math. Soc): primary 16N60; secondary 17C55, 17C6O.
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