integral element

Definitions

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.

Equivalents

Examples

“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 𝔅 containing the ring⟳ o of integral elements of the field k_0(π), 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 R) and if the element t of 𝔗 is integral over 𝔖, then t is also integral over R (that is, contained in 𝔖).”

“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.”

CEFR level