Abstract Witt ring

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There are many definitions of an abstract Witt ring. They all seek to define a class of rings that includes Witt rings of fields (of characteristic not two) and that is closed under fibre products, extensions by groups of exponent two and certain quotients. The need for such a class of rings became apparent early in the (still incomplete) classification of Noetherian Witt rings of fields.

Two series of definitions, that of J. Kleinstein and A. Rosenberg [a1] and M. Marshall [a2], led to the same class of rings, which is now the most widely used. In this sense, an abstract Witt ring is a pair $(R,G_R)$ where $R$ is a commutative ring with unit element $1$ and $G_R$ is a subgroup of the multiplicative group $R^*$ which has exponent two, contains $-1$ and generates $R$ additively. Let $I_R$ denote the ideal of $R$ generated by elements of the form $a+b$, with $a,b \in G_R$. It is further assumed that:

1) if $a \in G_R$, then $a \not\in I_R$;

2) if $a,b \in G_R$ and $a+b \in I_R^2$, then $a+b=0$;

3) if $a_1+\cdots+a_n = b_1+\cdots+b_n$, with $n \ge 3$ and all $a_i,b_i \in G_R$, then there exist $a,b,c_3,\cdots,c_n \in G_R$ such that $a_2+\cdots+a_n = a + c_3\cdots+c_n$ and $a_1 + a = b_1 + b$.

When $R$ is the Witt ring of a field $F$, then $G_R = F^*/(F^*)^2$ and property 3) is a consequence of the Witt cancellation theorem.


[a1] J. Kleinstein, A. Rosenberg, "Succinct and representational Witt rings" Pacific J. Math. , 86 (1980) pp. 99 – 137
[a2] M. Marshall, "Abstract Witt rings" , Queen's Univ. (1980)
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Abstract Witt ring. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by R.W. Fitzgerald (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article