I'm having a bit of confusion regarding the ideal in $\mathbb{R}[[x]]$ consisting of non-units and I'm probably making some silly mistake somewhere. It's clear from order considerations that the units of this ring are the non-zero constants and so my intuition has suggested that the ideal of non-units is principal and generated by $x$. But, in this case, every element of $(x)$ is divisible by $x$. However, $1+x\in \mathbb{R}$ is not divisible by $x$ yet it is non-unit. Can someone point out where my error is?
Thank you.