= Solution
{c}
The <Krull dimension> of a <commutative ring> $R$ is
$$
\boxed{\dim R=\sup\{n:\mathfrak p_0\subsetneq\mathfrak p_1\subsetneq\cdots\subsetneq\mathfrak p_n
\text{ is a chain of prime ideals of }R\}.}
$$
The length counts strict inclusions, so a chain with $n+1$ <prime ideals> has length $n$. The supremum may be infinite. A <field>, for example, has just the <prime ideal> $(0)$ and therefore has <Krull dimension> zero. For the zero <ring>, the <spectrum of a commutative ring> is empty; under the convention $\sup\varnothing=-\infty$, its dimension is $-\infty$.
Back to article page
