Mukai in his book "An introduction to invariants and moduli" uses a concept (first appearing section 2.3 finitely generated rings) which is known to him as algebraic ring extension (maybe more precisely it should be called algebraic domain extension). Unfortunately he never defines what he means by this notion. For example:
Lemma 2.22 Suppose that an integral domain $B$ is algebraic and finitely generated over a subring $A\subset B$. Then there exists a non-zero element $a\in A$ such that $B[a^{-1}]$ is integral over $A[a^{-1}]$.
I did a little bit of digging but I did not come up with a clear cut definition of algebraic ring extension anywhere. Algebraic field extension on the other hand is very well known. However with a little bit of thought I came with a few possibilities for the definition of algebraic ring extensions, and want to see which one is the true definition:
Possibility I: Let $A\subset B$ be an extension of integral domains. Let $K,L$ be respectively the field of fractions of $A,B$. Then $K\subset L$, an induced field extension. We call the ring extension $A\subset B$ algebraic if the induced field extension $K\subset L$ is algebraic.
The above possibility yields a weaker version, which turns out to be equivalent to the above:
Possibility I': Let $A\subset B$ be an extension of integral domains. We call this ring extension algebraic if every $x\in B$ satisfies a polynomial (not necessarily monomial) equation $a_0x^d+\cdots+a_d=0$ with $a_0, \cdots, a_d\in A$.
Then comes the last possibility which is based on the following observation:
Suppose $A$ is an integral domain, $K$ its field of fractions and $L/K$ an algebraic field extension. Then we have $A\subset K\subset L$. Every element $x\in L$ satisfies a polynomial equation (is a root of) $a_0x^d+\cdots+a_d=0$ with $a_0, \cdots, a_d\in A$. Define $B$ as the integral closure of $A$ in $L$. Then one checks that $L$ is the field of fractions of $B$.
This might result in a different possibility (in fact in the Lemma above, and as far as I know, whenever Mukai uses algebraic ring extension, the reason he does it is because he needs the above observation to hold):
Possibility II: Let $A\subset B$ be an extension of integral domains. Let $K,L$ be respectively the field of fractions of $A,B$. Then we call the ring extension $A\subset B$ algebraic if $B$ is the integral closure of $A$ in $L$.
Note that this is a stronger version of Possibility I'. This is because the above implies $B$ is integral over $A$. Hence $A\subset B$ is also algebraic in the sense of Possibility I'. Since Possibility I' is equivalent to Possibility I, Possibility II is in fact the strongest definition of them all.
