The standard definition of a ring is an abelian group that is a monoid under multiplication (with distributivity). However there are some books that have a weaker definition in that a ring only has to be closed under multiplication (no identity).
There is a problem in my algebra book asking me to prove that if a ring (defined in the second way) has $p$ elements, where $p$ is prime. If the multiplication is not trivial (i.e. sending everything to 0) then the ring is forced to have a multiplicative identity.
Its seems like a trivial proof, but I just can't see what I'm missing.
What I have so far: Given $R$ is an ring with p elements R is an abelian group of prime order, therefore it is cyclically generated, and of characteristic $p$ and isomorphic to $Z/pZ$. Essentially it boils down to showing $Z/pZ$ is forced to have a multiplicative identity, but I just can't see where this comes from (every resource I found seems to take this as fact). Since this is a requirement regardless of multiplicative structure I can't just use the fact that $Z/pZ - 0$ is a group under the typical multiplication.