Automorphism group of a polynomial ring

Definition

Let $R$ be a commutative unital ring, and $n$ be a natural number. The automorphism group of the polynomial ring in $n$ variables is defined as the group of permutations of the ring $R[x_{1},x_{2},\dots ,x_{n}]$ that are $R$-algebra automorphisms: in other words, the automorphism must be a ring automorphism and it must fix all the scalar polynomials, i.e., all the elements of $R$. The group is denoted $\operatorname {Aut} _{R}(R[x_{1},x_{2},\dots ,x_{n}])$.

IAPS structure

Further information: IAPS of automorphism groups of polynomial rings

The automorphism groups of polynomial rings form an IAPS of groups, i.e., there is a natural injective homomorphism:

$\Phi _{m,n}:\operatorname {Aut} _{R}(R[x_{1},x_{2},\dots ,x_{m}])\times \operatorname {Aut} _{R}(R[x_{1},x_{2},\dots ,x_{n}])\to \operatorname {Aut} _{R}(R[x_{1},x_{2},\dots ,x_{m},x_{m+1},\dots ,x_{m+n}])$

satisfying the associativity condition.