Let A be a commutative ring with identity and let R be its nilradical. We define a module of said ring to be flat if, on taking an exact sequence of its modules, tensoring it with the said module is exact. A ring that is absolutely flat means that every module has this property.

I am trying to prove the following. A/R is absolutely flat if and only if every prime ideal is maximal in A.

I know the following results. A is absolutely flat if and only if every principle ideal is idempotent. Also, A is absolutely flat if and only if A(M) is a field where A(M) denotes the localisation of A at M where M is any maximal ideal. Also, since A(P) is a local ring (P prime ideal of A) it has a unique maximal ideal.

Thus A/R ab. flat iff (A/R)(M) is a field
iff A(M)/R(M) is a field
iff R(M) is a maximal ideal of A(M)

Then I got a little stuck. Any suggestions would be much appreciated.