integral element

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 ${\\displaystyle {\\mathfrak {B}}}$ containing the ring ${\\displaystyle o}$ of integral elements of the field ${\\displaystyle k_{0}(\\pi )}$, distinct from ${\\displaystyle {\\mathfrak {B}}}$, and not contained in any other subring of ${\\displaystyle {\\mathfrak {B}}}$ distinct from ${\\displaystyle {\\mathfrak {B}}}$, is called a maximal ring of the algebra ${\\displaystyle {\\mathfrak {B}}}$. 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 ${\\displaystyle {\\mathfrak {S}}}$ is the ring of integral elements in a commutative ring ${\\displaystyle {\\mathfrak {T}}}$ (over a subring ${\\displaystyle {\\mathfrak {R}}}$) and if the element ${\\displaystyle t}$ of ${\\displaystyle {\\mathfrak {T}}}$ is integral over ${\\displaystyle {\\mathfrak {S}}}$, then ${\\displaystyle t}$ is also integral over ${\\displaystyle {\\mathfrak {R}}}$ (that is, contained in ${\\displaystyle {\\mathfrak {S}}}$).

   • 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 ${\\displaystyle s}$ is said to be integral over ${\\displaystyle R}$.
• The ring ${\\displaystyle S}$ is also said to be integral over ${\\displaystyle R}$, and to be an integral extension of ${\\displaystyle R}$.
• The set of elements of ${\\displaystyle S}$ that are integral over ${\\displaystyle R}$ is called the integral closure of ${\\displaystyle R}$ in ${\\displaystyle S}$. It is a subring of ${\\displaystyle S}$ containing ${\\displaystyle R}$.
• If ${\\displaystyle R}$ and ${\\displaystyle S}$ are fields, then ${\\displaystyle s}$ 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