# Proving that a ring with some properties is commutative

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 $$k\in \{1,2,...,p-1 \}$$