What are the units of $\Bbb R[X]/(X^2)$?
It is clear that the elements of $\Bbb R[X]/(X^2)$ is of the form $aX+b$ mod $X^2$ where $a,b \in \Bbb R$. I have observed that if $b=0$ then $aX+b$ is a divisor of zero and consequently it is not a unit. But if $a=0$ then for all $b \ne 0$ we have $aX+b$ is a unit. So we are now left with the case where neither $a$ nor $b$ is zero. In this case if $aX+b$ is a unit of the above quotient ring then $\exists\ cX+d$ with $c,d \in \Bbb R$ such that $(aX+b)(cX+d) +(X^2) = 1 + (X^2)$. Which is possible if $ad+bc=0$ and $bd=1$.But at this stage I got stuck. How can I proceed from here to reach at the desired stage.
Please help me in this regard.
Thank you in advance.
EDIT $:$ I have found $c = - \frac {a} {b^2}$ and $d = \frac {1} {b}$ since $b \ne 0$. So the set of units are $(\Bbb R[X] / (X^2)) \setminus (X)$. Isn't it?
Consider $aX+b\in \Bbb{R}[X]/(X^2)$. One has if $b\neq 0$
$$(aX+b)\left(-{a\over b^2}X+{1\over b}\right)=1$$
So any element $aX+b$ of $\Bbb{R}[X]/(X^2)$ with $b\neq 0$ is a unit