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In mathematics, **Lazard's universal ring** is a ring introduced by Michel Lazard in Lazard over which the universal commutative one-dimensional formal group law is defined.

In mathematics, **Lazard's universal ring** is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group law is defined.

There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let

*F*(*x*,*y*)

be

*x*+*y*+ Σ*c*_{i,j}*x**y*

for indeterminates

*c*_{i,j},

and we define the universal ring *R* to be the commutative ring generated by the elements *c*_{i,j}, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring *R* has the following universal property:

- For every commutative ring
*S*, one-dimensional formal group laws over*S*correspond to ring homomorphisms from*R*to*S*.

The commutative ring *R* constructed above is known as **Lazard's universal ring**. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degree 1, 2, 3, ... (where *c*_{i,j} has degree (*i* + *j* − 1)). Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so that *c* _{i, j} has degree *2(i+j-1)*, because the coefficient ring of complex cobordism is evenly graded.

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