If R is a commutative ring with identity that has exactly three ideals 0, I and R. Show that if a does not belong to the I then a is unit R.
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Note that R is a local ring. If you have seen that term before, you can apply some of the corresponding properties. – Kenneth Goodenough Oct 20 '20 at 21:25
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Since R is ring and 0 I and R are ideals of R. Let a € R-I. i. e a is not equal to zero as 0 € I. Now wo take maximal ideal J generated by a . i. e J = aR this is an ideal and subset of R. Since 1€R implies 1.a € J. Its means J not equal to I. Bcz a does not belong to the I. Also J not equal to zero bcz a not equal to zero. Therefore we must hav e J =R. Implies 1 €J so for some ring r €R. ra =1. This implies a is unit – Anjum butt Oct 21 '20 at 07:19