Extensions of primes, flatness, and intersection flatness

We study when R → S R \to S has the property that prime ideals of R"> encoding="application/x-tex">R extend to or unit ideal encoding="application/x-tex">S , and situation where this continues hold after adjoining same indeterminates both rings. prove if is reduced, every maximal contains only finitely many minimal primes left-bracket X 1 comma ellipsis Subscript n Baseline right-bracket"> stretchy="false">[ X 1 , …<!-- … <mml:mi>n stretchy="false">] encoding="application/x-tex">R[{X}_1, \, \ldots , {X}_{n}] S encoding="application/x-tex">S[{X}_1, for all alttext="n"> encoding="application/x-tex">n then flat over . give a counterexample flatness reduced quasilocal ring with infinitely by constructing non-flat -module M"> M encoding="application/x-tex">M such M equals P = P encoding="application/x-tex">M = PM P"> encoding="application/x-tex">P notion intersection use it in certain graded cases suffices examine just one closed fiber stable extension property.