nLab quotient ring

Contents

Contents

Idea

The quotient of a ring by an ideal.

Definition

Quotient by two-sided ideals

Given a ring RR and a two-sided ideal II with canonical RR-RR-bimodule monomorphism i:IRi:I \hookrightarrow R, the quotient of RR by II is the initial two-sided RR-algebra R/IR/I with canonical ring homomorphism h:RR/Ih:R \to R/I such that for every element aIa \in I, h(i(a))=0h(i(a)) = 0: for any other RR-algebra SS with canonical ring homomorphism k:RSk:R \to S such that for every element aIa \in I, k(i(a))=0 Sk(i(a)) = 0_S, there is a unique ring homomorphism l:R/ISl:R/I \to S such that lh=kl \circ h = k.

Quotient by left ideals

Given a ring RR and a left ideal II with canonical left RR-module monomorphism i:IRi:I \hookrightarrow R, the quotient of RR by II is the initial left RR-algebra R/IR/I with canonical ring homomorphism h:RR/Ih:R \to R/I such that for every element aIa \in I, h(i(a))=0h(i(a)) = 0: for any other RR-algebra SS with canonical ring homomorphism k:RSk:R \to S such that for every element aIa \in I, k(i(a))=0 Sk(i(a)) = 0_S, there is a unique ring homomorphism l:R/ISl:R/I \to S such that lh=kl \circ h = k.

Quotient by right ideals

Given a ring RR and a right ideal II with canonical right RR-module monomorphism i:IRi:I \hookrightarrow R, the quotient of RR by II is the initial right RR-algebra R/IR/I with canonical ring homomorphism h:RR/Ih:R \to R/I such that for every element aIa \in I, h(i(a))=0h(i(a)) = 0: for any other RR-algebra SS with canonical ring homomorphism k:RSk:R \to S such that for every element aIa \in I, k(i(a))=0 Sk(i(a)) = 0_S, there is a unique ring homomorphism l:R/ISl:R/I \to S such that lh=kl \circ h = k.

References

See also

Last revised on May 26, 2022 at 14:25:02. See the history of this page for a list of all contributions to it.