Ali, Majid M.; Smith, David J. Finite and infinite collections of multiplication modules. (English) Zbl 0987.13001 Beitr. Algebra Geom. 42, No. 2, 557-573 (2001). Summary: All rings are commutative with identity and all modules are unitary. In this note we give some properties of a finite collection of submodules such that the sum of any two distinct members is multiplication, generalizing those which characterize arithmetical rings. Using these properties we are able to give a concise proof of Patrick Smith’s theorem [cf. Arch. Math. 50, No. 3, 223-235 (1988; Zbl 0615.13003); theorem 8] stating conditions ensuring that the sum and intersection of a finite collection of multiplication submodules is a multiplication module. We give necessary and sufficient conditions for the intersection of a collection (not necessarily finite) of multiplication modules to be a multiplication module, generalizing Smith’s result. We also give sufficient conditions on the sum and intersection of a collection (not necessarily finite) for them to be multiplication. We apply D. D. Anderson’s characterization of multiplication modules [cf. Commun. Algebra 28, No. 5, 2577-2583 (2000; Zbl 0965.13003); theorem 1] to investigate the residual of multiplication modules. Cited in 6 Documents MSC: 13A05 Divisibility and factorizations in commutative rings 13C05 Structure, classification theorems for modules and ideals in commutative rings 13A15 Ideals and multiplicative ideal theory in commutative rings Keywords:arithmetical rings; multiplication modules; residual Citations:Zbl 0615.13003; Zbl 0965.13003 PDFBibTeX XMLCite \textit{M. M. Ali} and \textit{D. J. Smith}, Beitr. Algebra Geom. 42, No. 2, 557--573 (2001; Zbl 0987.13001) Full Text: EuDML EMIS