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"Since p is the additive identity in S, r∗p=p" Can you please explain why. I understand if p is the identity then r+p=r but not that statement.

Let S={p,q,r} and S=(S,+,*) a ring with identity. Let p be the identity for + and q the identity for *. Use the equation r*(r+q)=r*r+r*q to deduce that r*r=q. Attempt of a solution r*r=r*(r+q)- r*q =r*r+r*q - r*q But I'm not finding a clever way to deduce what is required. Any type of help...