Primary ideals in Witt rings

Generalizations of Primary Ideals in Commutative Rings

Let R be a commutative ring with identity. Let φ : I(R) → I(R) ∪ {∅} be a function where I(R) denotes the set of all ideals of R. A proper ideal Q of R is called φ-primary if whenever a, b ∈ R, ab ∈ Q−φ(Q) implies that either a ∈ Q or b ∈ √ Q. So if we take φ∅(Q) = ∅ (resp., φ0(Q) = 0), a φ-primary ideal is primary (resp., weakly primary). In this paper we study the properties of several genera...

Witt Rings and Matroids

The study of Witt rings of formally real fields in the algebraic theory of quadratic forms has led to a particularly good understanding of the finitely generated torsion free Witt rings. In this paper, we work primarily with a somewhat more general class of rings which can be completely characterized by (binary) matroids. The different types of standard constructions and invariants coming from ...

Characterizing Reduced Witt Rings Ii

Witt rings of quadratically presentable fields

This paper introduces an approach to the axiomatic theory of quadratic forms based on {tmem{presentable}} partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of {tmem{quadratically p...

Isotropy and Factorization in Reduced Witt Rings

We consider reduced Witt rings of finite chain length. We show there is a bound, in terms of the chain length and maximal signature, on the dimension of anisotropic, totally indefinite forms. From this we get the ascending chain condition on principal ideals and hence factorization of forms into products of irreducible forms. 2000 Mathematics Subject Classification: 11E81, 12D15