Generalized Pure Modules

Abstract

For right modules M<N over a ring R, consider any system of equations in M of the form ∑{xirij∣i∈I}=dj∈M,j∈J, where rij∈R. The usual definition of M as pure in N is that for any such a finite system, if the system is solvable in the bigger module N, then it is already solvable in M. Here the above ordinary concept of purity will be generalized by allowing I and J to be of possibly infinite cardinalities |I|<μ and |J|<ℵ for fixed cardinals μ and ℵ. In this way, generalized (μ<,ℵ<)-pure and absolutely pure concepts are defined in terms of μ and ℵ and studied. Here the number ℵ of relations of a module is simultaneously studied with the more familiar number μ of generators.