principal ideal in nLab

Contents

Contents

Definition

For rings

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 R -module/right sub- R R -module/sub- R R - 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.

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 I∈R/R ×I \in R/R^\times.

For lattices

A principal ideal in a lattice LL is an ideal II generated by an element x∈Rx \in R.

See also

• Bézout ring

• principal ideal domain

• principal ideal ring

• principal anti-ideal