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Quotient ring

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Summary

The ring R/IR/I of cosets of a two-sided ideal II, with addition and multiplication induced from RR; every ideal of a commutative ring is two-sided.

Record metadata

Canonical name
Quotient ring
Concept ID
quotient_ring
Content version
2.3.0
Primary field
Mathematics
Fields
Mathematics
Primary domain
Algebra
Domains
Algebra, Number theory, Algebraic geometry
Alternate terminology
No alternate terminology is currently authored for this record.
Concept type
Ideal quotient

Concept sources

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Commutative ringQuotient ring

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
algebra_commutative_ring_to_quotient_ring
Relation type
Quotient construction quotient-construction
Direction
source β†’ target
Endpoint roles
source: Quotients to; target: Obtained as quotient of
Authored annotation
identify elements modulo an ideal

Authored explanation

For an ideal Iβ—ƒRI\triangleleft R, the quotient identifies a,b∈Ra,b\in R exactly when aβˆ’b∈Ia-b\in I; the cosets a+Ia+I inherit well-defined ring operations.

How to interpret this relation type

Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.

Relation sources

IdealQuotient ring

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID
algebra_ideal_to_quotient_ring
Relation type
Mathematical formulation mathematical-formulation
Direction
source β†’ target
Endpoint roles
source: Mathematically formulates; target: Mathematically formulated using
Authored annotation
supplies the congruence

Authored explanation

The ideal determines the equivalence relation used in R/IR/I, and ideal absorption is what makes multiplication of cosets independent of representative.

How to interpret this relation type

A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.

Relation sources

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