Endomorphism algebras of modules with distinguished partially ordered submodules over commutative rings

AbstractLet R be a commutative ring and I = (I, ≤) be a partially ordered set. The paper is concerned with R1-modules M = M, M 1 | I), where M is an R-module with distinguished submodules M1 such that M1 ⊆ M1 for all i ≤ j ϵ I. We regard the endomorphism algebra End M = { ϕ ϵ End M | U 1 ϕ ⊆ U 1 for all i ϵ I and it will be the aim to derive a characterization of all those orderings (I, ≤) such that for every free R-module M1-infinite rank there is an R1-module structure M = (M, M1 | i ϵ I) on M with End M ≊ R. The orderings of interest are exactly the posets of infinite representation type derived by Kleiner. Moreover, the given result is extended to arbitrary R-algebras and to ‘rigid systems’ of R1-modules