nLab principal ideal

Contents

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 xRx \in R, or equivalently a left sub- R R -module/right sub- R R -module/sub- R R - R R -bimodule generated by xx.

This means there exists an element xIx \in I such that yy is a multiple of xx whenever yIy \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

Last revised on May 25, 2022 at 13:13:06. See the history of this page for a list of all contributions to it.