Principal ideal

An ideal (of a ring, algebra, semi-group, or lattice) generated by one element $\alpha$, i.e. the smallest ideal $L(\alpha)$ containing the element $\alpha$.

Rings

The left principal ideal $L(\alpha)$ of a ring $A$ contains, in addition to the element $\alpha$ itself, also all the elements

$$k\alpha+n\alpha$$

the right principal ideal $R(\alpha)$ contains all the elements

$$\alpha k+n\alpha$$

and the two-sided principal ideal $J(\alpha)$ contains all elements of the form

$$n\alpha+t\alpha+\alpha s+\sum _{ i }{ [(k_{ i }\alpha)l_{ i }+k_{ i }^\prime (\alpha l_{ i }^\prime )] } $$

where $k,t,s,k_{i},k_{i}^{\prime},l_{i},l_{i}^{\prime}$ are arbitrary elements of $K$ and $n\alpha=\alpha+\dots +\alpha$ ($n$ terms, $n\in\Z$). If $A$ is a ring with a unit element, in particular, for an algebra $A$ over a field, the term $n\alpha$ may be omitted: $$ L(a)=A\alpha,\qquad R(a)=\alpha A,\qquad J(\alpha)=A\alpha A\ . $$

Comments

Let $A$ be an integral domain with field of fractions $K$. A principal fractional ideal of $A$ is an $A$-submodule of $K$ of the form $Ar$ for some $r\in K$.

Semigroups

In a semi-group $S$ one also has left, right and two-sided ideals generated by an element $\alpha$, and they are equal, respectively, to

$$L(\alpha)=S^1\alpha,\qquad R(\alpha)=\alpha S^1,\qquad L(\alpha)=S^1\alpha S^1,$$

where $S^1$ is the semi-group coinciding with $S$ if $S$ contains a unit, and is otherwise obtained from $S$ by external adjunction of a unit.

Lattices

The principal ideal of a lattice $L$ generated by an element $\alpha$ is identical with the set of all $x$ such that $x \le \alpha$; it is usually denoted by $\alpha^\Delta$,$[{\alpha}]$, or $[{0,\alpha}]$ if the lattice has a zero. Thus,

$$\alpha^{ \Delta }=\alpha L=\{ { \alpha x:x\in L }\} .$$

In a lattice of finite length all ideals are principal.

Comments

Let $L$ be a lattice. Dual to the principal ideal generated by $\alpha \in L$ one has the principal dual ideal or principal filter determined by $\alpha$, which is the set $[\alpha)=\left\{ y\in L:\alpha \le y \right\} $. The principal ideal in $L$ determined by $\alpha$ is also denoted (more accurately) by $[\alpha)$.

A partially ordered set is a complete lattice if and only if it has a zero and every ideal in $L$ is principal.

References

[Be]	L. Beran, "Orthomodular lattices", Reidel (1985) pp. 4ff
[Gr]	G. Grätzer, "Lattice theory", Freeman (1971)
[Ku]	A.G. Kurosh, "Lectures on general algebra", Chelsea (1963) pp. 78; 86; 162 (Translated from Russian)
[Ly]	E.S. Lyapin, "Semigroups", Amer. Math. Soc. (1963) pp. Chapt. IV, Sect. 3 (Translated from Russian)