For a ring with identity, we denote by the Jacobson radical of the group of units of the ring of matrices with entries from and the group of units of respectively.
Here we proved this basic result that if is a finite field, then
In the same post, we also showed that if is a finite ring with identity, then
We are now going to use to find for any finite commutative ring with identity
Theorem. Let be a finite commutative ring with identity, and let be the set of all maximal ideals of Let Then
Proof (Y. Sharifi). Let By the Chinese Remainder Theorem,
and hence
The above ring isomorphism gives the group isomorphism
and so
On the other hand, by Example 8 in this post, and so, by applied to and also by
Example. Let be the prime factorization of an integer Then
Proof. Let The maximal ideals of are and
Therefore Also and so The result now follows from the Theorem.
Note. The above Example was posted on the Art of Problem Solving website as a problem yesterday; you can see the problem and my solution here.
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