Extensions of primes, flatness, and intersection flatness

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

An analogue of Vosper's theorem for extension fields

We are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which \begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath} when dimFS, dimFT ⩾ 2 and dimFST ⩽ [L : F] − 2.

Quantifier elimination in separably closed fields of finite imperfectness degree

The theory of separably closed fields of a fixed characteristic and a fixed imperfectness degree is clearly recursively axiomatizable. Ershov [1] showed that it is complete, and therefore decidable. Later it became clear that this theory also has the prime extension property in a suitable language (cf. [4, Proposition 1]); hence it admits quantifier elimination. The purpose of this work is to give an explicit, primitive recursive procedure for such quantifier elimination in the case of a finite imperfectness degree.To be precise, the language ∧ that we have in mind is the first order language of fields enriched with (m + 1)-place function symbols , where m = 0,1,2,… and 1 ≤ j ≤ pm. To interpret in a field M of characteristic p, consider the p-adic expansion of j – 1, and for x1,…,xm Є M let . If x1, …, xm) are p-independent and y Є M is p-dependent on them, then are linearly independent over Mp and y is linearly dependent on them. In this case there are unique such that define . Set otherwise.Denote by SCF(p,e) the theory of separably closed fields of characteristic p and finite imperfectness degree e, containing the above interpretation of the functions .