在交换代数中，可以根据整闭包的有限性将整环分成数类。以下均假设 为一整环。

注：一个代数簇的局部环或其完备化称作几何环，但此概念并不流行。

凡拟优环皆为永田环，所以代数几何中处理的环几乎都是永田环。是诺特整环而非永田环的例子首先由秋月康夫于1935年给出。

In commutative algebra, an integral domain A is called an N−1 ring if its integral closure in its quotient field is a finitely generated A module. It is called a Japanese ring (or an N−2 ring) if for every finite extension L of its quotient field K, the integral closure of A in L is a finitely generated A module (or equivalently a finite A-–algebra). A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, (or a pseudo–geometric ring) if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N−2 rings.) A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring (Danilov 2001), but this concept is not used much.