Finite commutative rings — Sylow components as ideals

Consider the following statements: (I) A p-sylow subgroup of the underlying additive group of a finite commutative ring R with unity is an ideal of R. (II) If M is a maximal ideal of a commutative ring R and R* be the group of units of R, such that R is a disjoint union of M and R*, then R has a maximal ideal other than M.