Let $R$ be a non-commutative ring with identity such that the identity map is the only ring automorphism of $R$. Prove that the set $N$ of all nilpotent elements of $R$ is an ideal of $R$.
In such a ring, every invertible element is central, else there is a nontrivial inner automorphism.
If $x$ is nilpotent, then $1-x$ is invertible.
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