Math 2.0

Associative rings and algebras Theorem

Universal property of the quotient ring

Builds on Quotient ringRing homomorphismKernel of a ring homomorphism

Statement

Let RR be a ring, let II be a two-sided ideal of RR and let π ⁣:R→R/I\pi \colon R \to R/I be the quotient map. Let f ⁣:R→Sf \colon R \to S be a ring homomorphism with I⊆ker⁡fI \subseteq \ker f. Then there is exactly one ring homomorphism fˉ ⁣:R/I→S\bar{f} \colon R/I \to S such that fˉ∘π=f\bar{f} \circ \pi = f. It is given by fˉ(a+I)=f(a)\bar{f}(a + I) = f(a).

Remarks

The map fˉ\bar{f} is well defined: if a+I=b+Ia + I = b + I, then a−b∈I⊆ker⁡fa - b \in I \subseteq \ker f, so f(a)=f(b)f(a) = f(b). The image of fˉ\bar{f} is the image of ff, and ker⁡fˉ=(ker⁡f)/I\ker \bar{f} = (\ker f)/I. With I=ker⁡fI = \ker f, this gives the first isomorphism theorem.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionIdeal, definition, no evidence levelIdealdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionKernel of a ring homomorphism, definition, no evidence levelKernel of a ring homo…definitionQuotient ring, definition, no evidence levelQuotient ringdefinitionUniversal property of the quotient ring, statement, no evidence levelUniversal property of…statement
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.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Ideal definition
  4. Ring homomorphism definition
  5. Two-sided ideal definition
  6. Kernel of a ring homomorphism definition
  7. Quotient ring definition
  8. Universal property of the quotient ring statement

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quotient_ring, Properties
About this page
Tag0R74MZYNRB
URIm2://0R74MZYNRB
Kindstatement, theorem
WritingW0 stub
MSC16D25
StatusThis copy of the site has no build data. Run m2 status 0R74MZYNRB.
UsesQuotient ring, Ring homomorphism, Kernel of a ring homomorphism

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