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.

This paper experimentally explored the influence of the wave spectrum shape variation on breakwater design. The energy spectrum function generally considered for the design of coastal structures is the JONSWAP spectrum. The laboratory results were therefore used to assess the impact of changing the spectrum shape parameter (PEF). We analysed armour stability and wave overtopping in a wave flume with a geometric similarity ratio of 1:30. The experimental results showed that the PEF has maximum influence on overtopping and wave pressures on the crown wall. For a PEF value of 3.3, overtopping was much higher (30% to 100% higher) than with a PEF of 1. Pressure on the crown wall was 20% higher with a PEF of 3.3 in comparison with that for a PEF equal to 1. The stability of the breakwater’s block armour is less sensitive to the PEF variation.

Let R be a commutative ring with identity. A proper ideal P is said to be weakly prime ideal of R if for every 0 ̸ = ab ∈ P where a, b ∈ R, implies a ∈ P or b ∈ P. The notion of weakly prime ideal was introduced by Anderson et al. in [Weakly prime ideals, Houston J. Math., 2003] as a generalization of prime ideals. The purpose of this paper is to study the form of weakly prime ideals of amalgamation of A with B along J with respect to f (denoted by A ◃▹ f J), introduced and studied by D'Anna et al. in [Amalgamated algebras along an ideal, Commutative Algebra and Its Applications, 2009]. Our results provide new techniques for the construction of new original examples of weakly prime ideals. Furthermore, as an application of our results, we provide an upper bound for the weakly Krull dimension of amalgamation.

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In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring along an ideal. Our results generate examples that enrich the current literature with new and original families of rings satisfying these properties.