Equations in Modules and Chain Conditions
by
John Dauns
Tulane University

For right modules M < N over a ring R, consider any system of equations in M of the form \sum{x_ir_ij | i in I}=d_j in M,  j in J, where r_ij in R. Then M is pure in N if any finite such a system, with |I| < \infty,  |J| < \infty, that is solvable in the bigger module N, is already solvable in the smaller module M. Here this latter concept of purity will be extended by allowing I and J to be either finite or infinite with cardinalities |I| < \mu and |J| < \aleph for fixed finite or mostly infinite cardinals \mu and \aleph. In this way, generalized (\mu ^< , \aleph ^< )-pure, absolutely pure, pure projective, and other similar concepts are defined in terms of \mu and \mu, and studied. Every module can be defined as the quotient of a \mu-generated free module modulo an \aleph-generated submodule. Here the number of relations |J|=\aleph of a module is simultaeaneously studied with the more familiar number |I|=\mu of generators.

Date received: March 18, 1999