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

  • Thread starter Thread starter Icebreaker
  • Start date Start date
  • Tags Tags
    Binary
Click For Summary

Homework Help Overview

The discussion revolves around the definition of multiplication in the ring R = \mathbb{Z}_5 \times \mathbb{Z}_5, particularly focusing on how to compute the product of two elements in this structure. Participants are exploring the implications of this definition and its relation to other algebraic structures.

Discussion Character

  • Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the pointwise definition of multiplication in R and its potential extension to other rings. There is a query regarding finding an isomorphism between \mathbb{Z}_5 \times \mathbb{Z}_5 and \mathbb{Z}_5[x]/(x^2 + 1), with some uncertainty about redefining multiplication in \mathbb{Z}_5 \times \mathbb{Z}_5.

Discussion Status

The conversation is ongoing, with participants providing insights into the definitions and relationships between different algebraic structures. Some guidance has been offered regarding the nature of product rings, but there is no explicit consensus on the approach to finding the isomorphism.

Contextual Notes

There is a mention of the potential need to redefine multiplication in \mathbb{Z}_5 \times \mathbb{Z}_5 to facilitate the exploration of isomorphisms, which raises questions about the assumptions underlying the original definitions.

Icebreaker
How is multiplication in [tex]R=\mathbb{Z}_5 \times \mathbb{Z}_5[/tex] defined? if (a,b) and (c,d) is in R, what's (a,b)(c,d)? (ac,bd)?
 
Physics news on Phys.org
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.
 

Similar threads

Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
  • · Replies 2 ·
Replies
2
Views
2K
Prove ##a\cdot b = b \cdot a ##using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
  • · Replies 9 ·
Replies
9
Views
3K
Prove ##a + b = b + a## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
[Complex analysis] Contradiction in the definition of a branch
  • · Replies 8 ·
Replies
8
Views
4K
Confusion about definition of a splitting field
  • · Replies 1 ·
Replies
1
Views
3K