Radical factorization for trivial extensions and amalgamated duplication rings

Let A subset of B be a commutative ring extension such that B is a trivial extension of A (denoted by A proportional to E) or an amalgamated duplication of A along some ideal of A (denoted by A?I). This paper examines the transfer of AM-ring, N-ring, SSP-ring and SP-ring between A and B. 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 B x C by a C-module E, where B is an SSP-ring, C a von Neumann regular ring and E 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.