Ring and Module Homomorphisms Coincide Only On The Identity
Theorem
Let \(R\) be a ring, and consider \(R\) as a module over \(R\) in the usual way.
Then \(\varphi : R \to R\) is both a ring and module homomorphism if and only if \(\varphi\) is the identity.
Proof
Suppose \(\varphi\) is a ring and module homomorphism. Then
\[ \varphi(rr') = \varphi(r)\varphi(r') = r\varphi(r').\]
As such, with \(r' = 1\) we have that \(\varphi(1) = 1\) because \(\varphi\) is a ring homomorphism and thus
\[ \varphi(r) = r.\]
On the converse, if \(\varphi\) is the identity then it trivially satisfies all the requirements of both a ring and module homomorphism.