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In trying to understand another questions answer(to a question I asked), I realized that my fundamental lack of knowledge was in regards to the following question:

In terms of functions, what does a quotient ring mean?

When we have a quotient ring of a polynomial ring: $$\Bbb R[x]/\langle f \rangle$$

We are setting $f\equiv 0\pmod f$

But when we have for example the ring of all continuous functions from the interval $[0,1]$ to $\Bbb R$. How do I think of a quotient ring of this?

ContinuityOfFailure
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  • Exactly the same. First, what is your ideal? It is generated by which functions? – Timbuc May 29 '15 at 12:07
  • Related:http://math.stackexchange.com/questions/299920/how-to-think-about-quotient-rings – Arpit Kansal May 29 '15 at 12:07
  • @Timbuc http://math.stackexchange.com/questions/1303818/maximal-ideals-of-the-ring-of-all-continuous-functions

    Is the related question and ideal. But I don't understand at all what it would mean to set that to be modulo $0$. That ideal is functions that are $0$ at a certain point

     – ContinuityOfFailure May 29 '15 at 12:10
  • Quotient rings arise from setting things in the ideal equal to $0$. So you're setting all of the functions vanishing at $a$ equal to the zero function. Maybe set the "setting stuff equal to zero" intuition aside for the time being and learn the real nuts and bolts of quotient rings - they partition the ring into cosets of an ideal which themselves behave nice under the ring's operations. What the elements of the ring are (numbers, functions, whatever) isn't necessarily important, the structure is. Alternatively - by the third isomorphism theorem, the quotient (over there) is just $\Bbb R$. – anon May 29 '15 at 12:10
  • @ContinuityOfFailure The ring consisting of functions does not substantially influence the core question about "what do quotient rings look like?," and again, there are probably at least a dozen previous posts addressing this. – rschwieb May 29 '15 at 15:13

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