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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

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    $\begingroup$ What do you mean by classification? And why are you interested in this? $\endgroup$
    – user6976
    Commented Mar 20, 2020 at 2:41
  • $\begingroup$ @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. $\endgroup$
    – Ali Taghavi
    Commented Mar 20, 2020 at 15:51
  • $\begingroup$ 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". $\endgroup$
    – user6976
    Commented Mar 20, 2020 at 17:53
  • $\begingroup$ @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? $\endgroup$
    – Ali Taghavi
    Commented Mar 20, 2020 at 18:11
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    $\begingroup$ 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, $\endgroup$
    – Salvatore Siciliano
    Commented Mar 20, 2020 at 18:55

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