Principal ideal

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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.