A <ring extension> $A\subseteq B$ is <module-finite ring extension>[finite] when $B$ is a <finitely generated module> over $A$. Every finite extension is integral by the <determinant trick>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 2 / a / ii / Solution

A <ring extension> $A\subseteq B$ is an <integral extension> when every $b\in B$ is an <integral element> over $A$: there is a <monic polynomial>
$$
b^d+a_{d-1}b^{d-1}+\cdots+a_0=0
$$
with all $a_i\in A$.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 2 / a / i / Solution

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A ring extension $A\subseteq B$ is an inclusion of rings that makes $B$ an $A$-algebra.


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