nLab principal ideal

Definition

For rings

A (left/right/2-sided) principal ideal in a ring $R$ is a left/right/2-sided ideal $I$ generated by an element $x \in R$, or equivalently a left sub-$R$-module/right sub-$R$-module/sub-$R$-$R$-bimodule generated by $x$.

This means there exists an element $x \in I$ such that $y$ is a multiple of $x$ whenever $y \in I$; we say that $I$ is generated by $x$. Thus every element $x$ generates a unique principal ideal, the set of all left/right/two-sided multiples of $x$: $a x$, $x b$, or $a x b$ if we are talking about left/right/two-sided ideals in a ring. Clearly, every ideal $I$ is a join over all the principal ideals $P_x$ generated by the elements $x$ of $I$.

In commutative rings

In commutative rings, since the set of all principal ideals is isomorphic to the quotient of (the multiplicative monoid structure on) $R$ by the group of units, a principal ideal $I$ is equivalently an element of the quotient monoid $I \in R/R^\times$.

For lattices

A principal ideal in a lattice $L$ is an ideal $I$ generated by an element $x \in R$.

See also

• Bézout ring

• principal ideal domain

• principal ideal ring

• principal anti-ideal