Math 2.0

Commutative algebra Definition

Annihilator

Builds on Commutative ringModule over a ringIdeal

Definition

Let RR be a commutative ring and let MM be an RR-module. The annihilator of MM is the set Ann⁡R(M)={r∈R:rm=0 for all m∈M}.\operatorname{Ann}_R(M) = \set{r \in R : rm = 0 \text{ for all } m \in M}. For an element m∈Mm \in M, the annihilator of mm is Ann⁡R(m)={r∈R:rm=0}\operatorname{Ann}_R(m) = \set{r \in R : rm = 0}.

Remarks

Ann⁡R(M)\operatorname{Ann}_R(M) and Ann⁡R(m)\operatorname{Ann}_R(m) are ideals of RR. Example: Ann⁡ℤ(ℤ/6ℤ)=6ℤ\operatorname{Ann}_\Z(\Z/6\Z) = 6\Z; multiplication by rr kills the class of 11 exactly when 6∣r6 \mid r, and then it kills every class. Non-example: 3∉Ann⁡ℤ(ℤ/6ℤ)3 \notin \operatorname{Ann}_\Z(\Z/6\Z), since 3⋅1=3≠03 \cdot 1 = 3 \ne 0 in ℤ/6ℤ\Z/6\Z. MM is a faithful module when Ann⁡R(M)=0\operatorname{Ann}_R(M) = 0. For an ideal JJ of RR, the annihilator of the module JJ is the ideal quotient (0:J)(0 : J). MM is naturally a module over R/Ann⁡R(M)R/\operatorname{Ann}_R(M).

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptModule over a ring, definition, no evidence levelModule over a ringdefinitionRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionIdeal, definition, no evidence levelIdealdefinitionAnnihilator, definition, no evidence levelAnnihilatordefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Group concept
  2. Module over a ring definition
  3. Ring definition
  4. Commutative ring definition
  5. Ideal definition
  6. Annihilator definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Annihilator_(ring_theory)
About this page
TagVQ31D68MDD
URIm2://VQ31D68MDD
Kinddefinition
WritingW0 stub
MSC13C99, 13A15
DefinesAnnihilator
UsesCommutative ring, Module over a ring, Ideal
Changed2 November 2026

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