Dictionary

integral element en · NOUN

Meanings

1. (qualifier:commutative algebra; ring theory; 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.
   • 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. 2004, Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko, Algebras, Rings and Modules, Volume 1, Kluwer Academic Publishers, page 209:
   • 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.

Forms

Relateds

integral extension · integral closure · algebraic integer · algebraic number

Translations (1)

it elemento intero (element of a given ring that is a root of some monic polynomial with coefficients in a given subring)