Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 3 / c / Solution

As an $A^{\prime }$-algebra, $A{\otimes }_R{A}^{\prime }$ is generated by the elements $a\otimes 1$. If $a$ obeys a monic relationover $R$, then $a\otimes 1$ obeys the same monic relation after applying the structure map $R\to A^{\prime }$. Thus every generator is an integral element. The subalgebra generated by finitely many integral elements is finite as a module, and therefore integral; each tensor involves only finitely many generators. Hence $A{\otimes }_R{A}^{\prime }$ is integral over $A^{\prime }$.

Solved by gpt-5.6-sol high.