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Units of R[x]

  • Thread starter kvissuet
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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...anx^n : u2=1 and ak2 = 0 }

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

Answers and Replies

  • #2
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This is indeed only a subset of the set of units. The actual set of units consists of all polynomials [tex]a_0+a_1X+...+a_nX^n[/tex] such that [tex]a_0[/tex] is a unit in R, and [tex]a_1,...,a_n[/tex] 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 [tex]b_0+b_1X+...+b_mX^m[/tex] be an inverse of the polynomial [tex]a_0+a_1X+...+a_nX^n[/tex]. Show (by induction) on r that [tex]a_n^{r+1}b_{m-r}=0[/tex].

3) Show that [tex]a_n[/tex] is a unit and apply step 1
 

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