integral element

English

Noun

integral element (plural integral elements)

1. (algebra, commutative algebra, ring theory) 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.
   • 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, American Mathematical Society 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.

   • 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 ${\mathfrak {S}}$).

   • 2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:

     In this paper the notion of the ring of all integral elements of a number field was put in the central place of his^{[Richard Dedekind's]} theory.

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 integral extension 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 algebraic element and the terms integral over and integral extension are replaced by algebraic over and algebraic extension (since the root of any polynomial is the root of a monic polynomial).

Translations

element of a given ring that is a root of some monic polynomial with coefficients in a given subring

• Italian: elemento intero m

See also

• algebraic integer
• algebraic number
• integral closure
• integral extension

Further reading

• Algebraic element on Wikipedia.Wikipedia
• Integral closure of an ideal on Wikipedia.Wikipedia
• Integral extension of a ring on Encyclopedia of Mathematics
• Algebraic extension on Encyclopedia of Mathematics
• Integral Extension on Wolfram MathWorld
• Algebraic Extension on Wolfram MathWorld