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.

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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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

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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Step 1: Proving A×B is abelian group

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Step 2: Proving A×B is abelian group

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Step 3: Multiplication is associative

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Step 4: Multiplication is distributive

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Step 5: When A×B is commutative

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Step 6: When A×B is ring with unity

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Step 7: Why is $Z_2×Z_3$ is commutative ring with unity

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