### Rings With Unity

The Ring ( R,  +,  .  ) is said to be a Ring with Unity if it contains an element denoted by 1  such that
a  .  1  =  1  .  a
if and only if     a    is element of    .

The unit element  1  is called multiplicative identity . It should not be confused with integer  1  though both are denoted by the both symbol .

In the ring of integers the unit element is the integer  1  whereas in the ring of matrices the unit element is the unit matrix of suitable order .  In the ring of all even integers there is no unit element and as such it is a Ring without Unity .

Similarly the function   e   denoted by    e(x) =  1   if and only if    x   is the element of    [ 0 , 1 ]  is the unit element because in this case  .

( ef ) x =  e(x)  f(x) = 1 . f(x) = f(x)
then fore   ef = f
Similarly , (fe) x = f(x) e(x) = f(x) .1  = f(x)
then fore   fe = f

We may also understand it by this discussion

Formally, a ring is an Abelian group (R, +), together with a second binary operation * such that for all a, b and c in R,
$a * (b*c) = (a*b) * c$
$a * (b+c) = (a*b) + (a*c)$
$(a+b) * c = (a*c) + (b*c)$
also, if there exists a multiplicative identity in the ring, that is, an element e such that for all a in R,
$a*e = e*a = a$
then it is said to be a ring with unity or a unitary ring. The unity is often denoted 1, since the number 1 is the unity in the common rings of numbers.
The ring in which e is equal to the additive identity must have only one element. This ring is called the trivial ring.
Rings that lie within other rings are called subrings. Maps between rings which respect the ring operations are called ring homomorphisms. Rings, together with ring homomorphisms, form a category (the category of rings). Closely related is the notion of ideals, certain subsets of rings which arise as kernels of homomorphisms and can serve to define factor rings. Basic facts about ideals, homomorphisms and factor rings are recorded in the isomorphism theorems and in the Chinese remainder theorem.

Noncommutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining noncommutative geometry based on noncommutative rings. Noncommutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an Abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of noncommutative rings are given by rings of square matrices or more generally by rings of endomorphisms of Abelian groups or modules, and by monoid rings.

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