Top (algebra)

In the context of a module "M" over a ring "R", the top of "M" is the largest semisimple quotient module of "M" if it exists. 
For finite-dimensional "k"-algebras ("k" a field) R, if rad("M") denotes the intersection of all proper maximal submodules of "M" (the radical of the module), then the top of "M" is "M"/rad("M"). In the case of local rings with maximal ideal "P", the top of "M" is "M"/"PM". In general if "R" is a semilocal ring (=semi-artinian ring), that is, if "R"/Rad("R") is an Artinian ring, where Rad("R") is the Jacobson radical of "R", then "M"/rad("M") is a semisimple module and is the top of "M". This includes the cases of local rings and finite dimensional algebras over fields.