Understanding Mappings between Quotient Rings

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Homework Help Overview

The discussion revolves around the concept of mappings between quotient rings, specifically the function p that maps elements from R/A to R/B. Participants are exploring the nature of these mappings and the implications regarding the existence of elements in the codomain prior to the mapping process.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions whether elements in R/B exist prior to the mapping and considers the implications of infinite domains and codomains on the mapping process. Some participants challenge this perspective, suggesting that mappings do not create or affect the existence of elements in the codomain.

Discussion Status

The discussion is active, with participants providing differing viewpoints on the nature of mappings and their implications. There is no explicit consensus, but some guidance is offered regarding the conceptual understanding of functions and mappings.

Contextual Notes

Participants are grappling with the definitions and properties of quotient rings and mappings, indicating a need for clarity on these foundational concepts. The discussion touches on the philosophical aspects of mathematical functions and their interpretations.

pivoxa15
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Homework Statement


If I was to map elements in R/A to R/B via the function p.

So p:R/A -> R/B

Can I assume there are no elements in R/B before the mapping?

Or is it more there are elements in R/B already before the mapping. However during the mapping, I highlight each element in R/B that was mapped by p from R/A. After the mapping has finished, the highlighted elements in R/B is the image of p. However if the domain and codomains are infinite then the mappings will never finish.
 
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What? There are exactly as many elements in R/B as there are elements in R/B. A 'mapping' does not suddenly 'make elements appear'. Functions are not processes that you switch on, wait for something to happen, and then it terminates like a machine in a factory (perhaps you're taking the metpahorical 'black box' description of functions too litereally).
 
So my last paragraph is correct?
 
No, your last paragraph doesn't make anysense either. Talking of maps as 'never finishing' doesn't make any sense at all. There is no time 'before' the map nor 'after' the map. This doesn't make any sense. R/A is a quotient ring, R/B is a quotient ring. The existence of maps between them does not affect thwm in the slightest.
 

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