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Commutative Ring

Ex.Let 12x≠q" style="WIDTH: 27pt; HEIGHT: 15pt" type="#_x0000_t75">  then 12Px, âˆھ, ∩ " style="WIDTH: 64.5pt; HEIGHT: 15pt" type="#_x0000_t75">  and 12Px, ∩, âˆھ" style="WIDTH: 66pt; HEIGHT: 15pt" type="#_x0000_t75">  are not rings. But 12Px, ∆, ∩" style="WIDTH: 63pt; HEIGHT: 15pt" type="#_x0000_t75"> from  a ring.

Ex(1). Let 12D= F:f:R→R" style="WIDTH: 86.25pt; HEIGHT: 15pt" type="#_x0000_t75">  define addition and multiple; on D by:-

12f+ga=fa+ga, a∈R " style="WIDTH: 178.5pt; HEIGHT: 15pt" type="#_x0000_t75"> 

12f.ga=fa.ga" style="WIDTH: 105pt; HEIGHT: 15pt" type="#_x0000_t75"> 

Then 12D, +, ." style="WIDTH: 37.5pt; HEIGHT: 15pt" type="#_x0000_t75">  is common; ring with identity the multi; identity element is the constant function whose value at 12x∈R" style="WIDTH: 26.25pt; HEIGHT: 15pt" type="#_x0000_t75">  is 1.

But 12D, +, ." style="WIDTH: 37.5pt; HEIGHT: 15pt" type="#_x0000_t75">  is not a ring, since the left dist; law

12f.g+ha≠f.g+f.h" style="WIDTH: 141pt; HEIGHT: 15pt" type="#_x0000_t75"> Doesn’t hold