For binary operation
* : A × A → A
e is called identity of * if
a * e = e * a = a
Here e is called identity element of binary operation.
Addition
+ : R × R → R
e is called identity of * if
a * e = e * a = a
i.e.
a + e = e + a = a
This is only possible if e = 0
Since a + 0 = 0 + a = a ∀ a ∈ R
0 is the identity element for addition on R
Multiplication
e is the identity of * if
a * e = e * a = a
i.e. a × e = e × a = a
This is possible if e = 1
Since a × 1 = 1 × a = a ∀ a ∈ R
1 is the identity element for multiplication on R
Subtraction
e is the identity of * if
a * e = e * a = a
i.e. a – e = e – a = a
There is no possible value of e where a – e = e – a
So, subtraction has no identity element in R
Division
e is the identity of * if
a * e = e * a = a
i.e. a/e = e/a = a
There is no possible value of e where a/e = e/a = a
So, division has no identity element in R *