Let A and B be rings. Define addition and multiplication on the Cartesian product A x B by                                         (x,y) + (x', y') = (x + x', y + y') and (x, y)(x', y') = (xx', yy').   Show that the A x B is also a ring using these axioms: (A) is an Abelian Group (B) Multiplication is associative (C) Multiplication is distributive over addition.   When is A x B commutative? When is it a ring with unity? Deduce from these facts that  Z_2 x Z_3 is a commutative ring with unity write down its addition and multiplication tables.

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