Math 2.0

Associative rings and algebras Theorem

First isomorphism theorem for rings

Builds on Quotient ringKernel of a ring homomorphismRing isomorphismUniversal property of the quotient ring

Statement

Let f ⁣:R→Sf \colon R \to S be a ring homomorphism. Then the kernel ker⁡f\ker f is a two-sided ideal of RR, and the image f(R)f(R) is a subring of SS. The map R/ker⁡f→f(R)R/\ker f \to f(R), a+ker⁡f↦f(a)a + \ker f \mapsto f(a), is a ring isomorphism. In particular, if ff is surjective, then R/ker⁡f≅SR/\ker f \cong S.

Remarks

The reduction map ℤ→ℤ/nℤ\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} has kernel nℤn\mathbb{Z}. The evaluation map ℝ[x]→ℂ\mathbb{R}[x] \to \mathbb{C}, p↦p(i)p \mapsto p(i), is surjective with kernel (x2+1)(x^2 + 1). So ℝ[x]/(x2+1)≅ℂ\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}. The theorem has the same form for groups and for modules.

Reading path

To understand this page from the ground up, read these 10 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…definitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionQuotient ring, definition, no evidence levelQuotient ringdefinitionUniversal property of the quotient ring, statement, no evidence levelUniversal property of…statementFirst isomorphism theorem for rings, statement, no evidence levelFirst isomorphism the…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. Ring isomorphism definition
  8. Quotient ring definition
  9. Universal property of the quotient ring statement
  10. First isomorphism theorem for rings statement

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Isomorphism_theorems, Theorem A (rings)
About this page
Tag0ZMYX7GMH0
URIm2://0ZMYX7GMH0
Kindstatement, theorem
WritingW0 stub
MSC16D25
StatusThis copy of the site has no build data. Run m2 status 0ZMYX7GMH0.
UsesQuotient ring, Kernel of a ring homomorphism, Ring isomorphism, Universal property of the quotient ring

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