PAIRS OF RINGS WITH A BIJECTIVE CORRESPONDENCE BETWEEN THE PRIME SPECTRA

Let A be a subring of a commutative ring B. If the natural mapping fro m the prime spectrum of B to the prime spectrum of A is injective (res pectively bijective) then the pair (A, B) is said to have the injectiv e (respectively bijective) Spec-map. We give necessary and sufficient conditions for a pair of rings A and B graded by a free abelian group to have the injective (respectively bijective) Spec-map. For this we f irst deal with the polynomial case. Let l be a field and k a subfield. Then the pair of polynomial rings (k[X], l[X]) has the injective Spec -map if and only if l is a purely inseparable extension of k.