Abstract
The objects of study in this paper are lattice ordered monoids. These are structures 〈L;⊕, 0L; ≤〉 where 〈L⊕, 0L〉 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.
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