Sums and products of elements integral over A remain integral over A. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
Objects and notation
An element \(b\) is integral over \(A\) when it satisfies a monic polynomial with coefficients in \(A\). An extension \(A\subseteq B\) is integral when every \(b\in B\) has this property.
A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.
Push the symbols
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
Structural reading
What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.
A hypothesis worth keeping
Algebraic over a fraction field and integral over the base ring are different conditions. Denominators make many algebraic elements nonintegral.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.