2015/08/06 by Gary F. Birkenmeier, Chris Ryan, Birkenmeier, Gary F. +2
Mathematics · #Advanced Topics in Algebra #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #math.RA #msc:16D70
paper · pdf · doi:10.48550/arxiv.1508.01543
arxiv created 2015/08/06 · arxiv updated 2015/08/10
In this paper we show that for a given set of pairwise comaximal ideals \Xi\i∈ I in a ring R with unity and any right R-module M with generating set Y and C(Xi)=∑k∈ℕ\underlineℓM(Xik), M=⊕i∈ IC(Xi) if and only if for every y∈ Y there exists a nonempty finite subset J⊆ I and positive integers kj such that \bigcapj∈ JXikj⊆\underlinerR(yR). We investigate this decomposition for a general class of modules. Our main theorem can be applied to a large class of rings including semilocal rings R with the Jacobson radical of R equal to the prime radical of R, left (or right) perfect rings, piecewise prime rings, and rings with ACC on ideals and satisfying the right AR property on ideals. This decomposition generalizes the decomposition of a torsion abelian group into a direct sum of its p-components. We also develop a torsion theory associated with sets of pairwise comaximal ideals.