Understanding Mappings between Quotient Rings

[pivoxa15]

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.

 

[matt grime]

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).

 

[pivoxa15]

So my last paragraph is correct?

 

[matt grime]

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.