On Summand Ideals in Commutative Reduced Rings#

Abstract

Summand intersection property (SIP) and summand sum property (SSP) of modules are studied in Garcia [Garcia, J. L. (1989). Properties of direct summands of modules. Comm. Algebra 17(1):73–92] and Wilson [Wilson (1986). Modules with the summand intersection property. Comm. Algebra 14(1):21–38] respectively, and these properties for C(X) are studied in Azarpanah [Azarpanah, F. (1999). Sum and intersection of summand ideals in C(X). Comm. Algebra 27(11): 5549–5560]. In this paper we give some topological characterizations of these properties in the case of a commutative reduced ring R. It is shown that R has SIP if and only if every intersection of clopen subsets of Spec(R) has a closed interior. This characterization shows that R has SIP whenever Spec(R) is locally connected or extremally disconnected. It is also shown that R has SSP if and only if Spec(R) has only finitely many components. If 𝒮 is a strong subspace of Spec(R), we can replace 𝒮 with Spec(R) and all of the above results remain valid. In semiprimitive Gelfand rings, SIP and SSP are equivalent.