Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract

Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M) = {PM | P ∈ Spec(R) and P ⊇ M ^⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.

Key Words:

• Multiplication module
• Prime submodule
• pm-module
• Spectral space
• Zariski topology

Acknowledgments

The authors are grateful for the referee's many valuable comments which have improved this paper. The first author and the second author are supported by the National Natural Science Foundation of China (1027105) and the first author and the third author are supported by the Doctorate Foundation of China Education Ministry (20020284009).

Notes

^#Communicated by I. Swanson.