Exercise 17, § 2

Let A be a commutative ring. Show that, in the algebra A[X] of polynomials in one indeterminate over A, the mapping (u, v) ↦ u(v) is a law of composition which is associative and left distributive with respect to addition and multiplication in A[X]. If A is an integral domain, show that the relation u(v) = 0 implies that u = 0 or that v is constant and that, if u ≠ 0 and the degree of v is > 0, the degree of u(v) is equal to the product of the degrees of u and v.