# Primes and units etc

• Feb 19th 2010, 11:52 AM
Primes and units etc
Let p be prime, m be a positive integer.

1-show that each element c of Z(p^m) is either a unit or is nilpotent.

I know nilpotent is when there is a pos. int. k so that c^k=0, and that a unit is such that c*k=1, but I don't know how to show this.

2- How many units are in Z(p^m)? nonunits? Explain.
• Feb 19th 2010, 03:04 PM
tonio
Quote:

Let p be prime, m be a positive integer.

1-show that each element c of Z(p^m) is either a unit or is nilpotent.

I know nilpotent is when there is a pos. int. k so that c^k=0, and that a unit is such that c*k=1, but I don't know how to show this.

2- How many units are in Z(p^m)? nonunits? Explain.

Hint for (1) and, after some thinking, for (2) as well: an element $c\in\mathbb{Z}_{p^m}$ is a unit iff $(c,p^m)=1\Longleftrightarrow p\nmid c \Longrightarrow$ the non-units are exactly the elements

of the form $c=ap^m\,,\,\,0\leq a, and now check what happens with a non-unit when you raise it to an appropiate power...

Tonio