An explicit ring quotient

Question

I need to calculate the quotient $\mathbb{Z}\left[\frac{1}{2}\right]/\left(3\mathbb{Z}\left[\frac{1}{2}\right]\right)$. I tried to factorize it as $\mathbb{Z}[X]/(2X-1,3)\cong\mathbb{F}_3[X]/(2X-1)$. I observed that in $\mathbb{F}_3[X]$, $(2X-1)=(2)(X-2)$ and that $\mathbb{F}_3[X]/(X-2)\cong \mathbb{F}_3$, $\mathbb{F}_3[X]/(2)\cong 0$, but the quotient seems to be quite a strange object and not as simple... My question is, is there a simple description of the quotient ring and how to demonstrate?

Answer 1

In $\Bbb F_3$, $2X-1=-X-1=-(X+1)$. Then your quotient is $\Bbb F_3[X]/(X+1)\cong\Bbb F_3$.