Are there rings whose multiplicative identity is not the number 1 or number 1-based? 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
 A: Consider $S = \{0, 2, 4, 6, 8\}$ with usual addition and multiplication modulo $10$. Then the identity element is $6$.
A: It depends on what exactly you mean by $1$. If you mean the real number $1$, then no an arbitrary ring has nothing to do with real numbers and hence nothing to do with the real $1$. However, $1$ is often understood to be defined as an object in a larger structure (here a ring) such that $1x = x1 = x$ for any $x$ in that structure, and this is exactly the definition of the multiplicative identity.
More important is the question of why we define the multiplicative identity to be so. The answer is that then multiplying by $1$ (whatever multiplying may mean) does not change anything. The next question is why we should have two identities, one for addition and one for multiplication, with distributivity of multiplication over addition. The answer is that addition represents some sort of combining while multiplication represents some sort of transforming (in general at least), where often $xy$ actually means "transform $y$ in a way specified by $x$", and so is not always commutative. It is very nice when the transformation commutes with combination, which is exactly the meaning of distributivity. Associativity is a natural part of the idea of combining and transforming. Additive inverse is nice because it corresponds to being able to undo a combination. Combinations of various kinds are also commutative. Thus we get all our ring axioms.
A: Consider the set $R$ of $2\times2$ matrices of the form
$$
R=\left\{\left(\begin{array}{cc}x&x\\x&x\end{array}\right)\big\vert\,x\in\Bbb{R}\right\}.
$$
The operations are the usual matrix multiplication and addition. I leave it to you to verify that $R$ is a ring, and that the matrix you get by setting $x=1/2$ is the multiplicative neutral element of $R$.
A: Although "multiply a ring element by another element of the same ring" and "multiply a ring element by an integer" are both called "multiplication", they are actually two distinct operations and should be viewed as such.
Multiplication of ring elements by integers may be defined according to the following:


*

*Multiplying a ring element by the integer zero yields the ring's additive identity.

*Multiplying a ring element by the integer one yields the same ring element.

*Multiplying a ring element by the whole number one yields the same ring element.

*For any integers x and y, multiplying a ring element by (x+y) is equivalent to multiplying by x, multiplying by y, and adding the results (using the ring's addition operator).


Some kinds of ring have a basis element such that any element of the ring will be an integer multiple of the multiplicative identity; in such cases, it may make sense to map integers to rings by saying that the integer N maps to N times the additive identity.  For such rings, the integer 1 will map to the additive identity, by definition.  For rings which do not have such a basis, ring elements may "look like" whole numbers, but will generally have no relation to them.
A: 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 $21\mathbb{Z}_{30}=\{0,3,6,9,12,15,18,21,24,27\}$.
A: Let $\Omega$ be an arbitrary non-empty set. Let $\mathfrak{P}(\Omega)$ denote the power set (set of all subsets) of $\Omega$.
The symmetric difference $A\Delta B$ for $A,B\subset\Omega$ is defined as follows:
$\qquad A\Delta B \equiv (A \backslash B) \cup (B \backslash A$).
Now $(\mathfrak{P}(\Omega), \Delta, \emptyset)$ is an abelian group and $(\mathfrak{P}(\Omega), \Delta, \emptyset, \cap, \Omega)$ is a commutative ring (with $\Delta$ as addition, $\emptyset$ as zero, $\cap$ as multiplication, and $\Omega$ as the unit of multiplication).
There are no "number-like" entities involved at all, everything is build using only the most basic set theory. The unit of multiplication is just a set. The set $\Omega$ could be anything. Families of subsets with certain properties that admit the above construction are used in measure theory and are called rings. However, the fact that these families are rings in the algebraic sense is not very useful, it's more like just a curious coincidence.
Moreover, the resulting ring is essentially the same as $(\mathbb{Z}/2\mathbb{Z})^{\Omega}$ with pointwise operations, so that $\Omega$ corresponds to the constant function $\Omega \mapsto 1$, so we get "the number $1$" (mod 2) again. The question by itself is not really meaningful, because one can always take any ring $\mathcal{R}$, call its unit of multiplication "1", and declare that $\mathcal{R}$ is just yet another kind of "numbers", so that your multiplicative unit becomes "number 1-based".
A: According to the kind explanation of @user21820, specifically this: "whatever multiplying may mean" I want also to make my own tryout of answer with a sample of algebra whose multiplicative identity is number i-based: 
For instance the family of algebras $\Bbb A_i$ whose multiplication function $*_i$ is declared as $a*_ib=\frac{a*b}{i}$. In this case the number $i$ will be the multiplicative identity of each $\Bbb A_i$ because $a*_ii= \frac{a*i}{i}=a$ 
A: Let $A$ be a Abelian group and let $R$ be the set of all homomorphisms from $A$ to itself.  Then $R$ is a ring under the operations of pointwise addition and function composition and the multiplicative identity is the identity mapping.
A: The ring $2\Bbb{Z}/10\Bbb{Z} = \{0, 2, 4, 6, 8\}$ has $6$ as its multiplicative identity.  There are many examples like this, but many authors/mathematicians tend to rule these out when talking about rings for a good reason:  the inclusion $2\Bbb{Z}/10\Bbb{Z} \hookrightarrow \Bbb{Z}/10\Bbb{Z}$ ought to be a ring homomorphism, mapping $1 \mapsto 1$.

Edit: to address comments and clarify.  I tend to to think of ring meaning unital ring and homomorphism to mean unital ring homomorphism.  In this way, they form a category, which is a nice way of thinking of rings and the ways that they map to one another in one cohesive context.  In many rich areas of algebra (representation theory of groups, Hopf algebras, quantum groups, etc.) the rings not only have units, but counits as well (more structure!)
However, there are situations in analysis where units are not available.  For example, $L^2(\Bbb{R})$, the space of functions $f: \Bbb{R} \to \Bbb{C}$ such that $\int_\Bbb{R} \lvert f(x) \rvert^2 \, dx < \infty$ does not have a multiplicative identity.  The constant function $1$ is not square-integrable.  And since the arabic numeral $1$ resembles the roman letter $i$ (especially uppercase $I$), these rings without multiplicative identity are sometimes called rngs.  :-)
