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Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition +2 and multiplication 2. Prove that the set R₁ x R2, with addition given by (11, 12) + (81, 82) := (1 +1 81, 12 +2 82) and multiplication given by (11, 12). (81, 82) := (r1·181, 12 2 82) is a ring. This ring is called the product of rings R₁ and R2.

Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition +2 and multiplication 2. Prove that the set R₁ x R2, with addition given by (11, 12) + (81, 82) := (1 +1 81, 12 +2 82) and multiplication given by (11, 12). (81, 82) := (r1·181, 12 2 82) is a ring. This ring is called the product of rings R₁ and R2.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.1: Definition Of A Ring
Problem 11E: Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the...
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Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition +2 and multiplication 2. Prove that the set R₁ x R2, with addition given by (11, 12) + (81, 82) := (1 +1 81, 12 +2 82) and multiplication given by (11, 12). (81, 82) := (r1·181, 12 2 82) is a ring. This ring is called the product of rings R₁ and R2.
Transcribed Image Text:Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition +2 and multiplication 2. Prove that the set R₁ x R2, with addition given by (11, 12) + (81, 82) := (1 +1 81, 12 +2 82) and multiplication given by (11, 12). (81, 82) := (r1·181, 12 2 82) is a ring. This ring is called the product of rings R₁ and R2.
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