If R is a commutative,Unitas ring and R[x] is a ring of polynomials in indeterminate x, then how do I know what the units in R[x] are?
I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.  Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.  Proof. Suppose $f(X)=a_{0}+a_{1}X+\cdots +a_{n}X^{n}$ is such that $a_{0}$ is a unit in $R$ and $a_{1},a_{2}, \dots,a_{r}$ are all nilpotent in $R$. Since $R$ is commutative, we get that $a_{1}X,a_{2}X^{2},\cdots,a_{n}X^{n}$ are all nilpotent and hence also their sum is nilpotent. Let $z = \sum a_{i}X^{i}$ then $a_{0}^{-1}z$ is nilpotent and so $1+a_{0}^{-1}z$ is a unit. Thus $f(X)=a_{0}+z=a_{0} \cdot (1+a_{0}^{-1}z)$ is a unit since product of two units in $R[X]$ is a unit.   I have not been able to get the converse part and would like to see the proof for the converse part.
My name includes space - How do I enter my username when typing command? My username in Minecraft is sun solar arrow. This name includes spaces. Situation - When I want to get OP privilege, a user who has OP privilege will enter: /op sun solar arrow  But my username includes spaces. So it shows an error message:  Invaild syntax  How do I enter my username? What's the 'correct' way of registering/installing an Assembly to the GAC? There seems to be lots of different ways to register assemblies with the GAC, as in, they 'work'.  However, what's the "proper" way of doing it?  In response to Lou Franco (and gacutil):  I'm using Gacutil for development, but it seems to me to be not the proper way to install it, since gacutil isn't included in the basic .NET utilities past .NET 1.1 - it's only a developer tool.  Additional: Gacutil (as seen in responses below) is not redistributable, and therefore should not be used in any app that you intend to give to people who are not developers.  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