Exercise 14, § 2

(a) Let M be a monoid whose law of composition is denoted by T; suppose further that M is totally ordered by an order relation x ≤ y such that the relations x < y and x' ≤ y', or x ≤ y and x' < y', imply x T x' < y T y'. Show that if A is an integral domain, the algebra over A of the monoid M has no divisors of 0.

(b) Deduce that if A is an integral domain, every polynomial algebra A[(X_i)_{i \in I}] is an integral domain.

(c) If A is an integral domain and f and g two polynomials in A[(X_i)_{i \in I}] such that fg is homogeneous and ≠0, show that f and g are homogeneous. In particular, the invertible elements of the ring A[(X_i)_{i \in I}] are the invertible elements of the ring A.