The Tropical Semiring is Local?

Question

Consider the Tropical semiring $\mathbb{T} = (\mathbb{R}\cup\{-\infty\},\min,+)$.

The Tropical semiring is also a local semiring? In other words, $\mathbb{T}$ has a unique maximal ideal?

Answer 1

Let $I$ be an ideal of $\mathbb{T}$. If $I$ contains a real $r$, then it contains every real $s$ since $s = r + (s-r)$. Moreover, it contains $-\infty$ since $r + (-\infty) = -\infty$. Thus $I = \mathbb{T}$. It follows that $I = \{-\infty\}$ is the unique proper ideal of $\mathbb{T}$. Indeed, it is closed under $\min$ and, for every $r \in \mathbb{T}$, $r + (-\infty) = -\infty$.