7. a) Recall for a commutative ring with unity, an ideal is maximal iff is an field. We will show that is a field.
Let be a non-zero element. To show has an inverse. Note that since be a non-zero element, . Define , Then note that , as . Hence we get
, hence going modulo we get , Hence is an unit. hence done the maximal ideal part.
b) Using the compactness of we will show that all maximal ideals of R is of the form , for some .
Proof by contradiction. Suppose there exists some maximal ideal which is not of the form for any . Then for all there exists be such that , note that since is continuous there exists an open neighbourhood of c, such that . Thus we get an open cover of , hence by compactness will get a finite sub cover say , and functions , such that . Consider . Note that as each , and for all , as if for some , , means we can find be such that , and , which is a contradiction. Hence there will not exists any maximal ideal other than .
Feel free to comment if you have any doubts. Cheers!
(7) Let R= {f [0,1] - R | f continuous} be the ring of all continuous...
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