Invariants of Lie superalgebras acting on associative algebras

Let R be a semiprime algebra over a field K acted on by a finite-dimensional Lie superalgebra L. The purpose of this paper is to prove a series of going-up results showing how the structure of the subalgebra of invariants R(L) is related to that of R. Combining several of our main results we have: Theorem: Let R be a semiprine K-algebra acted on by a finite-dimensional nilpotent Lie superalgebra L such that if characteristic K = p then L is restricted and if characteristic K = 0 then L acts on R as algebraic derivations adn algebraic superderivations. (i) If RL is right Noetherian, then R is a Noetherian right RL molecule. In particular, R is right Noetherian and is a finitely generated right RL-module. (ii) If RL is right Artinian, then R is an Artinian right RL-module. In particular, R is right Artinian and is a finitely generated right RL-module. (iii) If R(L) is finite-dimensional over K then R is also finite-dimensional over K. (iv) If R(L) has finite Goldie dimension as a right R(L)-module, then R has finite Goldie dimension as a right R-module. (v) R(L) has Krull dimension a as a right R(L)-module, then R has Krull dimension a as a right R(L)-module. Thus R has Krull dimension at most a as a right R-module. (vi) If R is prime and R(L) is central, then R satisfies a polynomial identity. (vii) If L is a Lie algebra and R(L) is central, then R satisfies a polynomial identity. We also provide counterexamples to many questions which arise in view of the results of this paper.