Formal group law

One-dimensional formal group law
Let $$R$$ be a commutative unital ring. A one-dimensional formal group law on $$R$$ is a formal power series $$F$$ in two variables, denoted $$x$$ and $$y$$, such that:

Condition (3) is redundant, i.e., it can be deduced from (1) and (2).

A one-dimensional commutative formal group law is a one-dimensional formal group law $$F$$ such that $$F(x,y) = F(y,x)$$. Two important examples of commutative formal group laws, that make sense for any ring, are the additive formal group law and the multiplicative formal group law.

Higher-dimensional formal group law
Let $$R$$ be a commutative unital ring. A $$n$$-dimensional formal group law is a collection of $$n$$ formal power series $$F_i$$ involving $$2n$$ variables $$(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n)$$ satisfying a bunch of conditions.

Before stating the conditions, we introduce some shorthand. Consider $$x = (x_1,x_2,\dots,x_n)$$ and $$y = (y_1,y_2,\dots,y_n)$$. Then, $$F(x,y)$$ is the $$n$$-tuple $$(F_1(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_2(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),\dots,F_n(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n))$$.

Condition (3) is redundant, i.e., it can be deduced from (1) and (2).

A commutative formal group law is a formal group law $$F$$ such that $$F(x,y) = F(y,x)$$. Two important examples of commutative formal group laws, that make sense for any ring, are the additive formal group law and the multiplicative formal group law.

For power series rings
A one-dimensional formal group law over a commutative unital ring $$R$$ gives a group structure on the maximal ideal $$ \langle t \rangle$$ in the ring $$Rt$$ of formal power series in one variable over $$R$$.

A one-dimensional formal group law can also be interpreted to give a group structure over the image of the maximal ideal $$\langle t \rangle$$ in any quotient ring of $$Rt$$; i.e., a ring of the form $$Rt/(t^n) \cong R[t]/(t^n)$$.

A $$n$$-dimensional formal group law over a commutative unital ring $$R$$ gives a group structure on the set of $$n$$-tuples of formal power series in one variable over $$R$$.

For arbitrary algebras over $$R$$
More generally, for any commutative $$R$$-algebra $$S$$, if $$N$$ is the set of nilpotent elements of $$S$$, then any $$n$$-dimensional formal group law $$F$$ over $$S$$ gives a group structure on the set $$N^n$$ of $$n$$-tuples over $$N$$. The formal group law thus gives a functor from the category of commutative $$R$$-algebras to the category of groups.

A particular case of this is when $$R$$ is a local ring and $$M$$ is its unique maximal ideal. In this case, we get what is called a $$R$$-standard group.