  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, Sch\"utzenberger
products, free products, Rees matrix semigroups, Brandt extensions,
Bruck-Reilly extensions and semilattices of semigroups.
