Top (mathematics)
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 R, if rad denotes the intersection of all proper maximal submodules of M, then the top of M is M/rad. 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, that is, if R/Rad is an Artinian ring, where Rad is the Jacobson radical of R, then M/rad is a semisimple module and is the top of M. This includes the cases of local rings and finite dimensional algebras over fields.
