On the set of integers let addition be defined in the usual way but define the \

Question

On the set of integers let addition be defined in the usual way but define the "product" of any two integers to be O. Is the set of all integers a ring with respect to addition and this new "multiplication"?

Explanation / Answer

as you have permitted upto addition, (Z,+) is an abelian group. given a, b are in Z ==> a.b = 0 is in Z and so, the multiplication '.' obeys closure law. a.(b.c) =a.(0) = 0 = 0.c = (a.b).c so, associativity holds. a(b+c) = 0 = 0+0 = a.0 + a.0 = a.b + a.c so, left distributivity holds in Z. (a+b)c = 0 = 0+ 0 = a.0+b.0 = a.c + b.c right distributivity holds in Z therefore, Z forms a ring under the given operations.

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