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
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.