Rings! Let I (not equal to R) be an ideal in a commutative ring R with identity.

Prove that R/I is an integral domain if and only if whenever ab is in I, either a is in I or b is in I.
Thanks!!!!!

Supose that R/I is an integral domain, and ab is in I for some a,b in I.
==> ab+I = 0+I
==> (a+I)(b+I) = 0+I
==> a+I = 0+I or b+I = 0+I, since R/I is an integral domain
==> a is in I or b is in I.

Conversely, suppose that (a+I)(b+I) = 0+I for some (a+I),(b+I) in R/I.
==> ab+I = 0+I
==> ab is in I
==> a is in I or b is in I by hypothesis
==> a+I = 0+I or b+I = 0+I.
Hence, R/I is an integral domain.

I hope this helps!