TITLE: Are there rings whose multiplicative identity is not the number 1 or number 1-based?
QUESTION [32 upvotes]: 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.
Idempotent Element
Abelian Group
Homomorphism
Identity Element
Subring

REPLY [3 votes]: Let $R$ be a commutative ring with unity.  Let $a$ be an idempotent element of $R$ (i.e., $a^2=a$), then the subring $aR=\{ar : r\in R \}$ has $a$ as its unity.
So for instance in $\mathbb{Z}_{30}$, $21$ is an idempotent element.  And so $21$ is the identity of $21\mathbb{Z}_{30}=\{0,3,6,9,12,15,18,21,24,27\}$.