  We call a semigroup $S$ weakly right noetherian if every right ideal of $S$
is finitely generated; equivalently, $S$ satisfies the ascending chain
condition on right ideals. We provide an equivalent formulation of the property
of being weakly right noetherian in terms of principal right ideals, and we
also characterise weakly right noetherian monoids in terms of their acts.
  We investigate the behaviour of the property of being weakly right noetherian
under quotients, subsemigroups and various semigroup-theoretic constructions.
In particular, we find necessary and sufficient conditions for the direct
product of two semigroups to be weakly right noetherian. We characterise weakly
right noetherian regular semigroups in terms of their idempotents. We also find
necessary and sufficient conditions for a strong semilattice of completely
simple semigroups to be weakly right noetherian. Finally, we prove that a
commutative semigroup $S$ with finitely many archimedean components is weakly
(right) noetherian if and only if $S/\mathcal{H}$ is finitely generated.
