University of MaraghehSahand Communications in Mathematical Analysis2322-580707120170701On the reducible $M$-ideals in Banach spaces27372387310.22130/scma.2017.23873ENSajadKhorshidvandpourDepartment of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.0000-0001-7824-9672AbdolmohammadAminpourDepartment of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.Journal Article20161119The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<\infty)$, then the number of reducible $M$-ideals does not exceed of $\frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The intersection of two $M$-ideals is not necessarily reducible. We construct a subset of the set of all $M$-ideals in a Banach space $X$ such that the intersection of any pair of it's elements is reducible. Also, some Banach spaces $X$ and $Y$ for which $K(X,Y)$ is not a reducible $M$-ideal in $L(X,Y)$, are presented. Finally, a weak version of reducible $M$-ideal called semi reducible $M$-ideal is introduced.https://scma.maragheh.ac.ir/article_23873_5d80e8659619ca4eed8588332a074319.pdf