integral element
التعريفات والمعاني
== English ==
=== Noun ===
integral element (plural integral elements)
(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
B
{\displaystyle {\mathfrak {B}}}
containing the ring
o
{\displaystyle o}
of integral elements of the field
k
0
(
π
)
{\displaystyle k_{0}(\pi )}
, distinct from
B
{\displaystyle {\mathfrak {B}}}
, and not contained in any other subring of
B
{\displaystyle {\mathfrak {B}}}
distinct from
B
{\displaystyle {\mathfrak {B}}}
, is called a maximal ring of the algebra
B
{\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
S
{\displaystyle {\mathfrak {S}}}
is the ring of integral elements in a commutative ring
T
{\displaystyle {\mathfrak {T}}}
(over a subring
R
{\displaystyle {\mathfrak {R}}}
) and if the element
t
{\displaystyle t}
of
T
{\displaystyle {\mathfrak {T}}}
is integral over
S
{\displaystyle {\mathfrak {S}}}
, then
t
{\displaystyle t}
is also integral over
R
{\displaystyle {\mathfrak {R}}}
(that is, contained in
S
{\displaystyle {\mathfrak {S}}}
).
==== Usage notes ====
Element
s
{\displaystyle s}
is said to be integral over
R
{\displaystyle R}
.
The ring
S
{\displaystyle S}
is also said to be integral over
R
{\displaystyle R}
, and to be an integral extension of
R
{\displaystyle R}
.
The set of elements of
S
{\displaystyle S}
that are integral over
R
{\displaystyle R}
is called the integral closure of
R
{\displaystyle R}
in
S
{\displaystyle S}
. It is a subring of
S
{\displaystyle S}
containing
R
{\displaystyle R}
.
If
R
{\displaystyle R}
and
S
{\displaystyle S}
are fields, then
s
{\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 ====
=== 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