If R is a domain prove that the only units in R x are units

If R is a domain, prove that the only units in R [x] are units in R. The domain Z_2 has only unit, Give an example of a infinite having only one unit. Let R be commutative ring and let

Solution

(i)

One direction is clear.

A unit in R is a unit in R[x].

Now suppose that f(x) is a unit in R[x]. Given a polynomial g, denote by d(g) the degree of g(x) (note that we are not claiming that R[x] is a Euclidean domain).

Now f(x)g(x) = 1.

Thus 0 = d(1)

= d(fg)

d(f) + d(g).

Thus both of f and g must have degree zero.

It follows that f(x) = f₀ and that f₀ is a unit in R[x].

(ii)

if R is an infinite ring ,then R has either infinitely many zero divisors ,or no zero divisors.