integral element

Noun

 * 1)  Given a commutative unital ring R with extension ring S (i.e., that is a subring of S), any element s ∈ S that is a root of some monic polynomial with coefficients in R.
 * 2) * 1956, Unnamed translator, D. K Faddeev, Simple Algebras Over a Field of Algebraic Functions of One Variable, in Five Papers on Logic Algebra, and Number Theory, Translations, Series 2, Volume 3, page 21,
 * A subring of $$\mathfrak B$$ containing the ring $$o$$ of integral elements of the field $$k_0(\pi)$$, distinct from $$\mathfrak B$$, and not contained in any other subring of $$\mathfrak B$$ distinct from $$\mathfrak B$$, is called a maximal ring of the algebra $$\mathfrak B$$. In a division algebra, the only maximal ring is the ring of integral elements.
 * 1) * 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, 2003 Softcover Reprint, page 172,
 * If $$\mathfrak S$$ is the ring of integral elements in a commutative ring $$\mathfrak T$$ (over a subring $$\mathfrak R$$) and if the element $$t$$ of $$\mathfrak T$$ is integral over $$\mathfrak S$$, then $$t$$ is also integral over $$\mathfrak R$$ (that is, contained in $$\mathfrak S$$).

Usage notes

 * Element $$s$$ is said to be integral over $$R$$.
 * The ring $$S$$ is also said to be integral over $$R$$, and to be an of $$R$$.
 * The set of elements of $$S$$ that are integral over $$R$$ is called the integral closure of $$R$$ in $$S$$. It is a subring of $$S$$ containing $$R$$.
 * If $$R$$ and $$S$$ are fields, then $$s$$ is called an and the terms integral over and integral extension are replaced by algebraic over and  (since the root of any polynomial is the root of a monic polynomial).

Translations

 * Italian: elemento intero