I have been studying the basics of integral closure and as much I have seen (from Atiyah's book) to prove that the integral elements of $A$ in $B$ form a ring we have to use modules and the fact that for $x$ to be an integral element it is enough for $A[x]$ to be a subring of a finitely generated $A$ module.
I was wandering if there exists another , more constructive way to do so, without using modules. To put it another way, if we know that for given $x,y$ there exists monic polynomials $p(X),q(Y)$ with coefficients in $A$ that $p(x)=p(y)=0$ then there exists a third monic polynomial with coefficients in $A$ $α(Χ)$ such that $α(x+y)=0$.
Milne provides two proofs of the claim here in his algebraic number theory notes. The first proof is on page $26$ and may be what you desire.
He writes:
"I shall give two proofs of this theorem. The first uses Newtons theory of symmetric polynomials and a result of Eisenstein, and the second is Dedekinds surprisingly modern proof, which avoids symmetric polynomials." (Where the latter does use module theory)