principal ideal (Rev #1) in nLab

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Definition

A principal ideal in a commutative ring RR is an ideal II if it is generated by a singleton. 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