Q. 54.7( 3 Votes )

Let A = Z × Z and

Answer :

Let, an element (e, f)Z × Z be the identity element, if


(a, b) * (e, f) = (a, b) = (e, f) * (a, b) (a, b) Z × Z


(af + be, bf) = (a, b) = (eb + fa, fb)


bf = b = fb … (1)


f = 1 … (2)


Also, af + be = a = eb + fa


As, f = 1


a + be = a


be = 0


e = 0


So, e = 0, f = 1


Hence, (0, 1) is the identity element.


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