nLab principal ideal




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.

In commutative rings

In commutative rings, since the set of all principal ideals is isomorphic to the quotient of (the multiplicative monoid structure on) RR by the group of units, a principal ideal II is equivalently an element of the quotient monoid IR/R ×I \in R/R^\times.

See also

Last revised on January 23, 2023 at 18:40:53. See the history of this page for a list of all contributions to it.