Multiplication in R=\mathbb{Z}_5 \times \mathbb{Z}_5: Operations and Definitions

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SUMMARY

Multiplication in the ring R=\mathbb{Z}_5 \times \mathbb{Z}_5 is defined pointwise, such that for elements (a,b) and (c,d) in R, the product is given by (a,b)(c,d) = (ac, bd). This operation can be extended to other rings, allowing for the definition of product rings G x H in a similar manner. The discussion also explores the challenge of finding an isomorphism between \mathbb{Z}_5 \times \mathbb{Z}_5 and \mathbb{Z}_5[x]/(x^2 + 1), emphasizing the need for a redefinition of multiplication in \mathbb{Z}_5 \times \mathbb{Z}_5 to facilitate this process.

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  • Understanding of ring theory and product rings
  • Familiarity with polynomial algebra, specifically in \mathbb{Z}_5
  • Knowledge of isomorphisms in algebraic structures
  • Basic concepts of pointwise operations in mathematical structures
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  • Research the properties of product rings, specifically G x H
  • Study the structure of \mathbb{Z}_5[x]/(x^2 + 1) and its relation to \mathbb{Z}_5 \times \mathbb{Z}_5
  • Learn about defining multiplication in different algebraic structures
  • Explore the concept of isomorphisms between rings and their applications
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Mathematicians, algebra students, and researchers interested in ring theory, polynomial algebra, and the study of isomorphisms in algebraic structures.

Icebreaker
How is multiplication in R=\mathbb{Z}_5 \times \mathbb{Z}_5 defined? if (a,b) and (c,d) is in R, what's (a,b)(c,d)? (ac,bd)?
 
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Usually pointwise (i.e. (a, b)(c, d) = (ac, bd) as you guessed). It can easily be extended to other groups (rings).
 
What do you mean by extending to other rings? I'm trying to find an isomorphism between Z5XZ5 and Z5[x]/X^2+1 and am having a hard time finding it. If I can redefine multiplication in Z5XZ5 then it will be easy.
 
I mean that given any rings G, H, you can easily define the product ring GxH in the same (pointwise) fashion.

I doubt the author (unless he or she said otherwise) intended for you to redefine Z_5 x Z_5.

Given any polynomial p, there are unique constants a, b and a polynomial q such that

p(x) = q(x)(x^2 + 1) + ax + b.

There seems to be an obvious function between Z_5[x]/(x^2 + 1) and Z_5 x Z_5 to try. But I haven't myself.
 

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