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In the mathematical discipline known as K-theory, the Milnor ring of a field F, named after John Milnor, is defined ^[1] as the graded associative ring $K_{*}^{M}(F)$ with unit, generated by symbols $\ell (a)$ (for $a\in F-\{0\}$ of degree one, with relations

\ell (ab)=\ell (a)+\ell (b)

,

\ell (a)\ell (1-a)=0.\,

One can show that $K_{0}^{M}(F)={\mathbb {Z} }$ , $K_{1}^{M}(F)=F-\{0\}$ .

The Milnor ring appears as one side of the Milnor conjecture.

References

^ T.A. Springer, A remark on the Milnor ring, Inventiones mathematicae, 1970

[1] T.A. Springer, A remark on the Milnor ring, Inventiones mathematicae, 1970

[1]

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 {{New unreviewed article|source=ArticleWizard|date=December 2009}}
-In K theory, '''Milnor ring''' of a field <math> F </math> is defined
+In the mathematical discipline known as [[K-theory]], the '''Milnor ring''' of a field ''F'', named after [[John Milnor]], is defined
-<ref> T.A.Springer, ''A remark on the Milnor ring'', Inventiones mathematicae, 1970 </ref>
+<ref> T.A. Springer, ''A remark on the Milnor ring'', Inventiones mathematicae, 1970 </ref>
-as the graded associative ring <math> K_*^M (F) </math> with unit, generated by symbols <math> \ell(a) </math> (for <math> a \in F -\{0\}</math> of degree one, with relations <br />
+as the graded associative ring <math> K_*^M (F) </math> with unit, generated by symbols <math> \ell(a) </math> (for <math> a \in F -\{0\}</math> of degree one, with relations
-<math> \ell(ab)=\ell(a)+\ell(b) </math>, <math> \ell(a)\ell(1-a)=0 </math>.
+: <math> \ell(ab)=\ell(a)+\ell(b) </math>, <math> \ell(a)\ell(1-a)=0. \, </math>
 One can show that <math> K_0^M (F)={\mathbb Z} </math>, <math> K_1^M (F)=F -\{0\} </math>.
 The Milnor ring appears as one side of the [[Milnor conjecture]].
 == References ==