Quasi-Baer module hulls and applications

• Published in: Journal of pure and applied algebra 2018-09, Vol.222 (9), p.2427-2455
• Main Authors: Lee, Gangyong, Park, Jae Keol, Tariq Rizvi, S., Roman, Cosmin S.
• Format: Article
• Language: English
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/j.jpaa.2017.09.021

Let V be a module with S=End(V). V is called a quasi-Baer module if for each ideal J of S, rV(J)=eV for some e2=e∈S. On the other hand, V is called a Rickart module if for each ϕ∈S, Ker(ϕ)=eV for some e2=e∈S. For a module N, the quasi-Baer module hull qB(N) (resp., the Rickart module hull Ric(N)) of N, if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull E(N) of N. In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R-module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let {Ki|i∈Λ} be any set of R-submodules of FR. For an R-module MR with AnnR(M)≠0, we show that MR⊕(⨁i∈ΛKi)R has a quasi-Baer module hull if and only if MR is semisimple. This quasi-Baer hull is explicitly described. An example such that MR⊕(⨁i∈ΛKi)R has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which N/t(N) is projective and AnnR(t(N))≠0, where t(N) is the torsion submodule of N, we show that the quasi-Baer hull qB(N) of N exists if and only if t(N) is semisimple. We prove that the Rickart module hull also exists for such modules N. Furthermore, we provide explicit constructions of qB(N) and Ric(N) and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum NR of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.