# When a finite codimensional subalgebra contains a finite codimension ideal?

What is a classification of all algebras $$A$$ (purely algebraic algebras, Banach or $$C^*$$ algebras or Lie algebras) with the following property:

Every finite codimensional subalgebra $$B$$ of $$A$$ contains a subalgebra $$I$$ such that $$I$$ is an ideal in $$A$$ and $$A/I$$ is a finite dimensional algebra.

One can ask a similar question for "Rings" but consider finite quotient rather than finite codimensionality in algebra case.

This question is inspired by this post and its comments conversation

• What do you mean by classification? And why are you interested in this? – user6976 Mar 20 at 2:41
• @MarkSapir The motivation for this question is the pure group theoric problem written in the attached link you find in this pist "every finit index subgroup contains a finit index normal sibgroup". By classification I mean some results as "An algebra has this property if ......" or " An algebra has this property if and only if it satisfies......". Or some examples or counter examples. – Ali Taghavi Mar 20 at 15:51
• What is a connection between the group theory problem and rings, $C*$-algebras and Lie algebras? An algebra has this property if it is finite or with 0-product (the product of any two elements is 0), or if every subalgebra of it is an ideal. A finite non-trivial field extension of an infinite field is a counterexample. There are lots of other similar trivial statements. I guess that is what you call "a classification". – user6976 Mar 20 at 17:53
• @MarkSapir I think you are collecting some (and not all ) sufficient conditions which looks trivial. A classification is a "iff" theorem. For example I think it is the case for commutative unital $C^*$ algebra. Do you have a counter example? Moreover what about Lie algebra case? – Ali Taghavi Mar 20 at 18:11
• In my opinion, a possible classification of Lie algebras with the required property is hopeless. However, it is worth mentioning that examples of infinite-dimensional simple Lie algebras containing subalgebras of finite codimension were constructed by Amayo in a paper published in the Proc. Lond. Math. Soc. in 1976. On the other hand, by a Theorem of Kukin, if a restricted Lie algebra $L$ over a field of characteristic $p>0$ contains a restricted subalgebra of finite codimension, then $L$ also contains a restricted ideal of finite codimension, – Salvatore Siciliano Mar 20 at 18:55