2016/04/08 by Helmut Zöschinger, Zöschinger, Helmut
Mathematics · #13C05 #13E15 #13F10 #16P20 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13C05 #msc:13E15 #msc:13F10 #msc:16P20
paper · pdf · doi:10.48550/arxiv.1604.02349
9 pages, in German
arxiv created 2016/04/08 · arxiv updated 2016/04/11
Let (R, \mathfrak m) be a commutative noetherian local ring. We investigate under which conditions an R-module M is generated by an ideal I, i.e. there exists an epimorphism I(Λ) \twoheadrightarrow M. If M is uniserial, i.e. L(M) is totally ordered and finite, this is equivalent to \mathfrakmn-1 ⋅ I \not⊂ AnnR(M) ⋅ I (length(M) = n ≥ 1). If M is cyclic and I = \mathfrakm, this is equivalent to: Either it is M ≅ R/\mathfrakp (R/\mathfrakp a discrete valuation ring) or M ≅ C/So(C) (C a uniserial R-module). If A is free and B is a submodule of A, then the Matlis dual (A/B)∘ = operatornameHomR(A/B, E) is I-generated if and only if B = (IB) :A I. In the case I = \mathfrakm, this condition leads to the "basically full ideals" considered by Heinzer, Ratliff~Jr. and Rush. By studying the dual condition M = I(M :X I) in the last section, we can generalize some results of that work.