Reading the basic definition of rings, I wondered if there are samples of rings whose multiplicative identity is not the number 1 or number 1-based (for instance the identity matrix is 1-based).
E.g. for $\Bbb Z$, if the definition of multiplication is modified (creating a non-standard algebra), could the multiplicative identity of the ring be another number, or the definition of multiplication must be "canonical" and must not be modified?
Is there a ring (currently in use for some field of Mathematics) sample of such non-1-based multiplicative identity?
I am learning by myself so I apologize if the question does not make much sense, thank you!
Update 2015/05/11: I will include some links to those wiki pages that were useful to understand the concepts written in the answers.