So, I was trying to find a counter-example that shows not every local ring's lattice of ideals is a chain. I think $F[[x_1,\cdots,x_n]]$ is a good counter-example but I'm not able to show that $F[[x_1,\cdots,x_n]]$ is a local ring. I read somewhere that $F[[x]]$ is indeed a local ring.
So here comes the question:
If $R$ is a local ring, what can we say about $R[[x]]$? Is it a local ring too?
I'm looking for a simple proof that doesn't use commutative algebra and localizations to show that. An elementary undergraduate level proof is appreciated.