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Commutative Ring1

Theorem:-

In any ring  if

Proof:

So by cancelation law

Similarly       obtain

Note:-

is called zero ring , 0 is identity of (+, .) the identity and zero element in the zero ring are equal.

Theorem:-

Let  be a ring with identity . Then the elements 0 and 1 are distinct  i.e. .

Proof:-

Since . If 1=0 then  contradiction

  are distinct.