Properties of group

There are two properties of group as follows
(a) Identity element in a group is unique.

 Proof : 
           Let e be the identity element of a group ( G, O ), Then we have to show that 'e ' is unique.
If possible let e' be an other identity element of the same group ( G, O ).

∵ e be the identity.
∴ a O e = e O a = a ∀ a ∊ G

If a = e'
then, e' O e = e O e' = e' ∀ e' ∊ G    ➖➖➖➖➖ (i)

Again, e' is the identity
So, a O e' = e' O a = a ∀ a ∊ G
e O e' = e' O e = e ∀ e ∊ G   ➖➖➖➖➖ (ii)

from (i) and (ii) we get,

 e = e'

Hence, there is one and only one identity element is present in G.




(b) Formation of Ring (Ring)
Ring :  A non-empty set 'R' with two binary operation '+'  ( called ordinary addition ) and '*' ( called ordinary multiplication ) is said to be a ring ( R, +, * ) if ( R, + ) is an abelian group, ( R, * ) is a semi-group  and '*' is distributed over '+' .


   In other words, a non-empty set 'R' with two binary operation '+' ( called ordinary addition ) and '*' ( called ordinary multiplication ) is said to be a ring (R, + , * ) if the following properties holds --
For binary operation '+' ( ordinary addition )

  ( i ) Clouser law : a, b ∊ R ⇒ (a + b) ∊ R,
 ( ii ) Associative law : a, b, c ∊ R ⇒ ( a + b ) + c = a + ( b + c ) ∊ R,
( iii ) Commutative law : a, b ∊ R ⇒ ( a + b ) = ( b + a ) ∊ R,
( iv ) Existence of identity element : There exists an element 0 ∊ R ( called zero element ), such that a + 0 = a ∀ a ∊ R.
( v ) Existence of inverse element : Corresponding to each element a ∊ R there exists an element 'x' a ∊ R called additive inverse, such that a + x  = 0.


For binary operation '*' ( ordinary multiplication )

 ( i ) Clouser law : a, b ∊ R ⇒ (a * b) ∊ R,
( ii ) Associative law : a, b, c ∊ R ⇒ ( a * b ) * c = a * ( b * c ) ∊ R,


For both binary operations '+' (ordinary addition ) and '*' ( ordinary multiplication )

( i ) Distributive law : a, b, c ∊ R ⇒ a * ( b  + c ) = a * b + a * c  ∊ R,  ( b  + c ) * a =  b * a + c * a  ∊ R



Example : The set of  'Z' of integers is a ring under ordinary addition and multiplication.

Ring with unity

   A ring ( R, + , * ) is called a ring with unity, if there exists an element e ∊ R such that a * e = e * a = a ∀ e ∊ R. Then, e is called unity element.
i.e. Identity element for multiplication exists in R.
If no such element exists in R, then we called it a ring without unity.

 Example : Set of even integers ( R, +, * )

 R = { ........, -200, ----, -4, -2, , 0 , 2, 4, ......, 200 ,...........}. ( Ring without unity ).

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