 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 101

For a prime ideal $\mathfrak p$, its height is the supremum of lengths $r$ of strict chains
$$
\mathfrak p_0\subsetneq\mathfrak p_1\subsetneq\cdots\subsetneq\mathfrak p_r=\mathfrak p.
$$
The <height of an ideal> $I$, without assuming $I$ prime, is
$$
\boxed{\operatorname{ht}(I)=
\inf_{\mathfrak p\supseteq I}\operatorname{ht}(\mathfrak p)}.
$$
The <Krull dimension> of a ring $R$ is
$$
\boxed{\dim R=\sup\{r:\mathfrak p_0\subsetneq\cdots\subsetneq\mathfrak p_r
\text{ are prime ideals of }R\}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 4 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 4 / i / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 4 / i / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)