Semigroups with right congruences of finite index

It is proved that all non-trivial right congruences of a semigroup $S$ have finite indices if and only if either $S$ is finite or $S$ is isomorphic to a subsemigroup of the additive group of integers with the adjoint zero. This result allows to describe the semigroup algebras whose non-trivial right ideals have the finite codimensions.