Rings which are almost polynomial rings

Abstract:

If A is a commutative ring with identity and B is a unitary A-algebra, B is locally polynomial over A provided that for every prime p of A, ${B_p} = B{ \otimes _A}{A_p}$ is a polynomial ring over ${A_p}$. For example, the ring $Z[\{ X/{p_i}\} _{i = 1}^\infty ]$, where $\{ {p_i}\} _{i = 1}^\infty$ is the set of all primes of Z, is locally polynomial over Z, but is not a polynomial ring over Z. If B is locally polynomial over A, the following results are obtained, B is faithfully flat over A. If A is an integral domain, so is B. If $\mathfrak {a}$ is any ideal of A, then $B/\mathfrak {a}B$ is locally polynomial over $A/\mathfrak {a}$. If p is any prime of A, then pB is a prime of B. If B is a Krull ring, so is A and the class group of B is isomorphic to the class group of A . If A is a Krull ring and B is contained in an affine domain over A, then B is a Krull ring. If A is a noetherian normal domain and B is contained in an affine ring over A, then B is a normal affine ring over A. If M is a module over a ring A, the content of an element x of M over A is defined to be the smallest ideal ${A_x}$ of A such that x is in ${A_x}M$. A module is said to be a content module over A if ${A_x}$ exists for every x in M. M is a content module over A if and only if arbitrary intersections of ideals of A extend to M. Projective modules are content modules. If B is locally polynomial over a Dedekind domain A, then B is a content module over A if and only if B is Krull.