Is the ideal $I = \{f \mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on the interval $[0, 1]$ a maximal ideal?
Is the ideal $I = \{f\mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on $[0, 1]$ a maximal ideal?
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abstract-algebra
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7Why do you ask? What do you think? What have you tried? – 2012-12-30
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0yes,infact I={f:f(c)=0} are the all possible maximal ideals – 2012-12-30
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6**Hint:** Try to determine the quotient $C[0,1]/I$ using the map $\phi: g\mapsto g(0)$. What is $\ker \phi$? What is ${\rm Im}~\phi$? – 2012-12-30
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0where is the compactness of [0,1] used? – 2012-12-30
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0Why do you think compactness would be used, @K.Ghosh? – 2012-12-30
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0in C(0,1) does this hold? i don't know – 2012-12-30
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5@K.Ghosh, Compactness is needed for the statement that every max ideal is of the form $I_c = \{f: f(c) = 0\}$, not this one. – 2012-12-30
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0This question is contained in http://math.stackexchange.com/questions/375400/maximal-ideals-in-the-ring-of-real-functions-on-0-1 – 2013-05-26