(adsbygoogle = window.adsbygoogle || []).push({}); Given:

Denote a ring G = { all functions from ℝ --> ℝ }

And set N = { all functions from ℝ --> ℝ such that for any x in ℝ, f(x) = 0 }

Want to prove:

To prove that N is the maximal ideal of G by showing that the quotient ring G / N is isomorphic to the set of real numbers ℝ

** What I'm still getting stuck at this point is the following:

1/ What should an element of G / N look like. Since G contains all real function, is it true that an element in G / N should be something like f(x) + N, where f(x) is in ℝ ? or is it something else ?

2/ My approach is to try to define a map, say f, from G/N to ℝ. But then I get stuck on what (in ℝ) should I send elements of G/N to ?

For first approach, I try to send everything in G/N to f(0). But then I recognize that by the mean of "all functions", G may have some functions which are not defined at 0, say f(x) = 1/x, so f(0) doesn't work.

For second try, I try to send everything in G/N to f(1). But again, the same issue as my first approach comes up again.

Would someone please help me on this problem?

Thanks in advance.

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# Show a set of functions such that f(x)=0 is maximal ideal of the set of all real func

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