\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
Summand sum property $(SSP)$ and summand intersection property $(SIP)$ of modules are studied in [8] and [15] respectively. In this paper we give some topological characterizations of these properties in $C(X)$. It is shown that the ring $C(X)$ has $SIP$ if and only if every interseciton
of closed-open subsets of $X$ has a closed interior. This characterization then shows that for a large class of topological spaces, such as locally connected spaces and extremally disconnected spaces, the ring $C(X)$ has $SIP$. It is also shown that $C(X)$ has $SSP$ if and only if the space $X$ has only finitely many components. Finally, using summand ideals of $C(X)$, we will give several algebraic characterizations of some disconnected spaces.
\end{document}