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A is a ring with the next properties:

a) the order of $1$ is p (prime) in the group $(A,+)$

b) there exists $B \subset A$ with $p$ elements such that : for all $x,y \in A$, exists $b \in B$ which verifies $xy=byx$.

Prove that A is commutative. Can somebody give me some tips, pelase? I have no idea how to solve it.

I managed to get that $k \cdot1$ is invertible for all $k\in \{1,2,...,p-1 \}$

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