The field of rational functions $\mathbb Q(x)$. Also described as the quotient field for $\mathbb Q[x]$. We establish an order on this field by declaring a nonzero element $\frac{f(x)}{g(x)}$ to be positive if the leading coefficients of $f$ and $g$ have the same sign. The element $x$ is larger than all of $\mathbb Q$, so that the field is non-Archimedean.
Keywords polynomial ring ring of quotients
| Name | Measure | |
|---|---|---|
| cardinality | $\aleph_0$ | |
| global dimension | left: 0 | right: 0 |
| Krull dimension (classical) | 0 | |
| weak global dimension | 0 |
| Name | Description |
|---|---|
| Idempotents | $\{0,1\}$ |
| Jacobson radical | $\{0\}$ |
| Left singular ideal | $\{0\}$ |
| Left socle | $R$ |
| Nilpotents | $\{0\}$ |
| Right singular ideal | $\{0\}$ |
| Right socle | $R$ |
| Units | $R\setminus\{0\}$ |
| Zero divisors | $\{0\}$ |