Radical factorization for trivial extensions and amalgamated duplication rings

Abstract

Let [Formula: see text] be a commutative ring extension such that [Formula: see text] is a trivial extension of [Formula: see text] (denoted by [Formula: see text]) or an amalgamated duplication of [Formula: see text] along some ideal of [Formula: see text] (denoted by [Formula: see text]. This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between [Formula: see text] and [Formula: see text]. We study the transfer of those properties to trivial ring extension. Call a special SSP-ring an SSP-ring of the following type: it is the trivial extension of [Formula: see text] by a C-module [Formula: see text], where [Formula: see text] is an SSP-ring, [Formula: see text] a von Neumann regular ring and [Formula: see text] a multiplication C-module. We show that every SSP-ring with finitely many minimal primes which is a trivial extension is in fact special. Furthermore, we study the transfer of the above properties to amalgamated duplication along an ideal with some extra hypothesis. Our results allows us to construct nontrivial and original examples of rings satisfying the above properties.