Units in R[x]: A Comprehensive Definition

[kvissuet]

Homework Statement

Define the set of units in R[x] where R is a commutative Ring and R[x] the polynomial ring.

Homework Equations

Unit: X is a unit in R if there exist a Y in R such that XY=1

The Attempt at a Solution

At first I thought it was this:
R[x]* = {u +a₁x + a₂x^2...a_nx^n : u²=1 and a_k² = 0 }

But I feel that this is just a subset of the actual set of units.

 

[micromass]

This is indeed only a subset of the set of units. The actual set of units consists of all polynomials a_0+a_1X+...+a_nX^n such that a_0 is a unit in R, and a_1,...,a_n are nilpotent elements of R. Here is a scheme that will help you prove this fact:

1) For a general ring A: if x is nilpotent, then 1+x is a unit. In fact, if u is a unit and if x is nilpotent, then u+x is a unit.

2) Let b_0+b_1X+...+b_mX^m be an inverse of the polynomial a_0+a_1X+...+a_nX^n. Show (by induction) on r that a_n^{r+1}b_{m-r}=0.

3) Show that a_n is a unit and apply step 1