×

Finite and infinite collections of multiplication modules. (English) Zbl 0987.13001

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.

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
PDFBibTeX XMLCite
Full Text: EuDML EMIS