Type dimensions of modules
by
John Dauns
Tulane University
Coauthors: Yiqiang Zhou

A complement submodule N <= M is a type submodule if for any C <= M with N \cap C=0, N \perp C, i.e. N and C have no nonzero isomorphic submodules. The type dimension of any module M is the finite or infinite cardinal number t.dim M=|I|, where I indexes a maximal pairwise orthogonal direct sum \oplus{A(i) | i in I} <= M of type submodules A(i) <= M with A(i) \perp A(j) for i =/= j in I. For an infinite cardinal \aleph >= \aleph₀, a module M satisfies the \aleph ^< -type ascending chain condition (\aleph ^< -t-acc.) if for any ordinal indexed chain of submodules X₀ < ... < X_i < ... <= M, 0 <= i < \sigma for some ordinal \sigma, we have t.dim \oplus{M/X_i | i < \sigma} < \aleph. For regular cardinals \aleph >= \aleph₀, rings R_R satisfying the \aleph ^< -t-acc. are investigated, and how this condition influences the structure of injective R-modules.

Date received: November 11, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-16.