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
the right principal ideal contains all the elements
and the two-sided principal ideal contains all elements of the form
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,
In a semi-group one also has left, right and two-sided ideals generated by an element , and they are equal, respectively, to
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,
In a lattice of finite length all ideals are principal.
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.
|[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