 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2011/algebra/3/a/solution

I

of a commutative ring is an additive subgroup such that

r x \in I

for every

r \in R

and

x \in I

A \neq = R

such that

x y \in A

implies

x \in A

y \in A

(x + A) (y + A) = 0 + A

in the quotient ring, then

x y \in A

. Primality implies

x + A = 0 + A

y + A = 0 + A

. Therefore there are no two nonzero cosets whose product is zero:

R / A has no nonzero zero-divisors .

Properness also ensures its identity coset is not zero, so this quotient is an integral domain.

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