Projectivity and linkage for completely join irreducible ideals of an expanded group

Abstract

Specialized from lattice theory to the lattice of ideals of an expanded group, the equivalence relation of projectivity between two intervals of ideals I[A, B] and I[C, D] of an expanded group V yields an isomorphism between B / A and D / C. Conversely, E. Aichinger has shown that if V is finite and B / A and D / C are minimal factors of V such that B / A and D / C are isomorphic as $C_0(V)$-modules where $C_0(V)$ is the nearring of functions on V that are 0-preserving and congruence preserving, then I[A, B] and I[C, D] are projective. Restricting this equivalence relation to intervals of the form $I[U^-,U]$ where U is a completely join irreducible ideal of V and $U^-$ is the unique ideal of V maximal in U plays a central role in this work. In this paper we introduce another equivalence relation involving the completely join irreducible ideals of V called linkage. We explore this linkage equivalence relation and its relationship to projectivity for completely join irreducible ideals. As a byproduct of our studies we will extend Aichinger’s result from the finite case to the one where $C_0(V)$ satisfies a finiteness condition called the weak descending chain condition on right ideals previously studied by the authors.