Structure Theorem for Finite Commutative Rings with unity

Question

The Structure Theorem for Finite Commutative Rings with unity state that:

A finite commutative ring $R$ with multiplication identity is isomorphic to a direct sum of local rings.

Suppose all the maximal ideals of $R$ are $M_1, M_2, ..., M_n$, and let $J$ be the Jacobson radical of $R$. There exists a positive integer $k$ such that $J^k=0$.

There is an assertion in the proof which says that $M_i$ is the unique maximal ideal such that $M_i^k\subseteq M_i$, why?

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The complete proof of the theorem see

• p. 9, lem. 9 in the thesis http://bfhaha.blogspot.tw/2014/06/thesis-classification-of-finite-rings.html
• p.95, thm. (VI.2) in the book "Finite rings with identity", Bernard R. McDonald
• p.40, thm. 3.1.4 in the article "Finite commutative rings and their applications", Gilberto Bini and Flamino Flamini

Answer 1

Because maximal ideals are prime, and $M^k\subseteq P$ implies $M\subseteq P$ for a prime ideal $P$. If then $M$ is maximal, $M=P$.