Question about ideals and integral domains.

Suppose that A is a ring and I is an ideal of A. Prove that the quotient ring A/I is an integral domain if and only if I satisfies the following:

and or .

I have tried it both ways using a contradictory argument but to no avail. Help much appreciated.

Re: Question about ideals and integral domains.

An ideal which satisfies the second condition is said to be prime. Use the fact that is the same thing as the class of modulo is the class of , and the product of two classes is the class of the product.

Re: Question about ideals and integral domains.

this is pretty basic: suppose I is a prime ideal of A. then if (x+I)(y+I) = I in A/I, and x is not in I, then xy + I = I, so xy is in I, and since I is prime, and x is not in I, y is in I.

but this means that y+I = I, that is, A/I has no zero divisors. provided that A was a commutative ring with unity in the first place, A/I is an integral domain (some authors do not require commutativity nor an unit).

(the condition I ≠ A ensures we have some other element besides I = 0 + I in A/I, so that we have non-zero divisors at all).

the converse is proven similarly: using a direct approach, rather than contradiction, works well.