I have some confusion about some properties of noetherian posets. I am defining a poset $X$ to be $\textbf{noetherian}$ if every ideal of $X$ Is finitely generated. I am trying to show that this is equivalent to the statement: $X$ has the descending chain condition (DCC) and no infinite anti chains.
Before I even prove this, though, I wanted to make sure it made sense to me, and I immediately came up with a counter example in my mind (of course this counter example is wrong, but I cannot see why! This is what I need help in).
Consider the poset $(\mathbb{R}^+,\leq)$. I believe every ideal of $\mathbb{R}^+$ is finitely generated. In fact, it seems that all ideals of $\mathbb{R}^+$ can be written as the interval $[a,\infty)$ , and thus all ideals are generated by a single principle ideal.
However, the DCC on $\mathbb{R}^+$ does not seem to hold. For if we take the sequence \begin{equation}\frac{1}{1},\frac{1}{2},\frac{1}{3},...\end{equation} It never stabilizes.
There must be something very wrong in my understanding, can someone help clarify what I did wrong here?
Thank you!