Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Algebra / 3 / a / Solution

An ideal $I$ of a commutative ring is an additive subgroup such that $rx\in I$ for every $r\in R$ and $x\in I$. A prime ideal is a proper ideal $A\ne R$ such that $xy\in A$ implies $x\in A$ or $y\in A$.

If $(x+A)(y+A)=0+A$ in the quotient ring, then $xy\in A$. Primality implies $x+A=0+A$ or $y+A=0+A$. Therefore there are no two nonzero cosets whose product is zero:Properness also ensures its identity coset is not zero, so this quotient is an integral domain.