Integral element

English

Noun

1. Template:Lb 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, Template:W Translations, Series 2, Volume 3, page 21,

     A subring of $𝔅$ containing the ring $o$ of integral elements of the field $k_0(\pi )$, distinct from $𝔅$, and not contained in any other subring of $𝔅$ distinct from $𝔅$, is called a maximal ring of the algebra $𝔅$. 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 $𝔖$ is the ring of integral elements in a commutative ring $𝔗$ (over a subring $\Re$) and if the element $t$ of $𝔗$ is integral over $𝔖$, then $t$ is also integral over $\Re$ (that is, contained in $𝔖$).

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

Translations

• Italian: Template:T

See also

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Further reading

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• Integral extension of a ring on Template:W
• Algebraic extension on Template:W
• Integral Extension on Template:W
• Algebraic Extension on Template:W