When a group algebra (semigroup algebra) is an Artinian algebra?

Question

When a group algebra (semigroup algebra) is an Artinian algebra?

We know that an Artinian algebra is an algebra that satisfies the descending chain condition on ideals. I think that a group algebra (semigroup algebra) is an Artinian algebra if the group algebra (semigroup algebra) satisfies the descending chain condition on ideals. Are there other equivalent conditions that determine when a group algebra (semigroup algebra) is an Artinian algebra? Thank you very much.

Answer 1

A result of E. Zelmanov (Zel'manov), Semigroup algebras with identities, (Russian) Sib. Mat. Zh. 18, 787-798 (1977):

Assume that $kS$ is right artinian. Then $S$ is a finite semigroup. The converse holds if $S$ is monoid.

See this assertion in Jan Okniński, Semigroup algebras.Pure and Applied Mathematics, 138 (1990), p.172, Th.23.

Answer 2

It is a result of [I.G. Connell, On the group ring, Canad. J. Math. 15 (1963), 650–685] that the group algebra $kG$ over a field $k$ of a group $G$ is left artinian iff the group $G$ is finite.