The ascending chain condition on principal right ideals for semigroup constructions

We call a semigroup $\mathcal{R}$-noetherian if it satisfies the ascending chain condition on principal right ideals, or, equivalently, the ascending chain condition on $\mathcal{R}$-classes. We investigate the behaviour of the property of being $\mathcal{R}$-noetherian under the following standard semigroup-theoretic constructions: semidirect products, Schützenberger products, free products, Rees matrix semigroups, Brandt extensions, Bruck-Reilly extensions and semilattices of semigroups.