Principal ideal
An ideal (of a ring, algebra, semi-group, or lattice) generated by one element , i.e. the smallest ideal containing the element
.
The left principal ideal of a ring
contains, in addition to the element
itself, also all the elements
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the right principal ideal contains all the elements
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and the two-sided principal ideal contains all elements of the form
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where are arbitrary elements of
and
(
terms,
). If
is a ring with a unit element, the term
may be omitted. In particular, for an algebra
over a field,
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In a semi-group one also has left, right and two-sided ideals generated by an element
, and they are equal, respectively, to
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where is the semi-group coinciding with
if
contains a unit, and is otherwise obtained from
by external adjunction of a unit.
The principal ideal of a lattice generated by an element
is identical with the set of all
such that
; it is usually denoted by
,
, or
if the lattice has a zero. Thus,
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In a lattice of finite length all ideals are principal.
Comments
Let be an integral domain with field of fractions
. A principal fractional ideal of
is an
-submodule of
of the form
for some
.
Let be a lattice. Dual to the principal ideal generated by
one has the principal dual ideal or principal filter determined by
, which is the set
. The principal ideal in
determined by
is also denoted (more accurately) by
.
A partially ordered set is a complete lattice if and only if it has a zero and every ideal in is principal.
References
[a1] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1963) pp. Chapt. IV, Sect. 3 (Translated from Russian) |
[a2] | L. Beran, "Orthomodular lattices" , Reidel (1985) pp. 4ff |
[a3] | G. Grätzer, "Lattice theory" , Freeman (1971) |
[a4] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) pp. 78; 86; 162 (Translated from Russian) |
Principal ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Principal_ideal&oldid=12133