Topological algebras with ascending or descending chain condition (Q1300324)

All algebras under consideration are real and unital. An algebra is said to be Noetherian (resp. Artinian) if the family of all its left ideals satisfies the ascending (resp. descending) chain condition. The following main results are proved in this paper. (1) Every commutative Noetherian F-algebra is a Q-algebra. (2) Let \(A\) be a unitary F-algebra which is a Q-algebra. Then \(A\) is a Noetherian algebra if and only if every left ideal of \(A\) is closed. (3) Let \(A\) be a commutative normed algebra. Then \(A\) is finite-dimensional if and only if \(A\) is an Artianian algebra. Some other related results are also given.