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 Z₂ x Z3 is acommutative ring with unity write down its addition and multiplication tables.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.2: Complex Numbers And Quaternions

Problem 50E

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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? D