[[Ideal]]
# Principal ideal

A (left/right/two-sided) [[ideal]] $I \trianglelefteq R$ is called **principal** iff it is generated by a single element. #m/def/ring 
Specifically, $I$ is a

- **left principal ideal** iff $I = Ra$ for some $a \in R$;
- **right principal ideal** iff $I = aR$ for some $a \in R$;
- **two-sided principal ideal** iff $I = RaR$ for some $a \in R$.

A ring in which every ideal is (left/right/two-sided) principal is called a (left/right/two-sided) [[principal ideal ring]].

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#state/tidy | #lang/en | #SemBr