Principal ideal

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In mathematics, specifically ring theory, a principal ideal is an ideal ${\displaystyle I}$ in a ring ${\displaystyle R}$ that is generated by a single element ${\displaystyle a}$ of ${\displaystyle R}$ through multiplication by every element of ${\displaystyle R.}$ The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset ${\displaystyle P}$ generated by a single element ${\displaystyle x\in P,}$ which is to say the set of all elements less than or equal to ${\displaystyle x}$ in ${\displaystyle P.}$

The remainder of this article addresses the ring-theoretic concept.