Rank ideals and Capelli identities

We define and study a family of completely prime rank ideals in the universal enveloping algebra . A rank ideal is a noncommutative analogue of a determinantal ideal, the defining ideal for the closure of the set of n×n matrices of fixed rank. We introduce a notion of rank for -modules and determine the rank of simple highest weight modules and of simple finite-dimensional modules. The main tools are Capelli-type identities and filtered algebra.