Decomposition of algebras with finite special rank (Q2748966)

An associative algebra \(A\) over a field \(F\) is said to be of finite special rank if there exists a natural number \(n\) such that any subalgebra of \(A\) which is finitely generated admits \(n\) generators.NEWLINENEWLINENEWLINEThe authors prove the following results: (1) Any algebra of finite special rank is algebraic. (2) If an algebra \(A\) over a finite field or an algebraically closed field possesses finite special rank, then \(A\) is locally finite dimensional. (3) Let \(A\) be an algebra over an algebraically closed field. If \(A\) is countably generated and has finite special rank, then it can be decomposed into a semidirect sum of its radical and a semisimple algebra.