View source for Principal ideal

An [[Ideal|ideal]] (of a ring, algebra, semi-group, or lattice) generated by one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747301.png" />, i.e. the smallest ideal containing the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747302.png" />.

The left principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747303.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747304.png" /> contains, in addition to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747305.png" /> itself, also all the elements

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747306.png" /></td> </tr></table>

the right principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747307.png" /> contains all the elements

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747308.png" /></td> </tr></table>

and the two-sided principal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p0747309.png" /> contains all elements of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473011.png" /> are arbitrary elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473014.png" /> terms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473015.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473016.png" /> is a ring with a unit element, the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473017.png" /> may be omitted. In particular, for an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473018.png" /> over a field,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473019.png" /></td> </tr></table>

In a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473020.png" /> one also has left, right and two-sided ideals generated by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473021.png" />, and they are equal, respectively, to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473023.png" /> is the semi-group coinciding with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473024.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473025.png" /> contains a unit, and is otherwise obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473026.png" /> by external adjunction of a unit.

The principal ideal of a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473027.png" /> generated by an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473028.png" /> is identical with the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473030.png" />; it is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473032.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473033.png" /> if the lattice has a zero. Thus,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473034.png" /></td> </tr></table>

In a lattice of finite length all ideals are principal.



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473035.png" /> be an integral domain with field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473036.png" />. A principal fractional ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473037.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473038.png" />-submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473039.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473040.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473041.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473042.png" /> be a lattice. Dual to the principal ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473043.png" /> one has the principal dual ideal or principal filter determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473044.png" />, which is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473045.png" />. The principal ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473046.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473047.png" /> is also denoted (more accurately) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473048.png" />.

A partially ordered set is a [[Complete lattice|complete lattice]] if and only if it has a zero and every ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074730/p07473049.png" /> is principal.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.S. Lyapin,   "Semigroups" , Amer. Math. Soc.  (1963)  pp. Chapt. IV, Sect. 3  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Beran,   "Orthomodular lattices" , Reidel  (1985)  pp. 4ff</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Grätzer,   "Lattice theory" , Freeman  (1971)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Kurosh,   "Lectures on general algebra" , Chelsea  (1963)  pp. 78; 86; 162  (Translated from Russian)</TD></TR></table>