1) Let I be an ideal in a ring R. Prove that every element in R/I is a solution

Question

1) Let I be an ideal in a ring R. Prove that every element in R/I is a solution of x^2 = x if and only if for every element a in R, element a^2 - a is in I.

2) Let I does not equal R be an ideal in a commutative ring R with identity. Prove that R/I is an integral domain if and only if whenever element ab is in I, either element a is in I or element b is in I.

Explanation / Answer

1) Consider any element in R/I its of the form a+I where a is in R now we want every element to satisfy x^2=x iff (a+I)(a+I) = (a^2 + I) = (a+I) iff a^2 - a belongs to I since a+I=b+I iff a-b belongs to I this is a good resource:http://en.wikipedia.org/wiki/Quotient_ring 2) As the ring is commutative so is the quotient ring, so only condition we need to check is that there should be no zero divisors ie if xy = 0 then either x or y is zero Now x,y are in the quotient ring so x = (a+I) , y = (b+I) hence xy=0 -> x=0 or y=0 means (a+I)(b+I) = (ab+I) = (0+I) -> a+I=I or b+I = I iff a lies in I or b lies in I ie ab lies in I iff a lies in I or b lies in I all the steps are iff so proved comment if you have any doubts

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