principal ideal (Rev #2) in nLab

Contents

Definition

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

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

See also

• Bézout ring

• principal ideal domain