Math 2.0

Associative rings and algebras Proposition

A ring homomorphism is injective iff its kernel is zero

Builds on Ring homomorphismKernel of a ring homomorphism

Statement

Let f ⁣:R→Sf \colon R \to S be a ring homomorphism. Then ff is injective if and only if its kernel is {0}\{0\}.

Remarks

The proof uses only the addition: f(a)=f(b)f(a) = f(b) holds if and only if f(a−b)=0f(a - b) = 0, that is, a−b∈ker⁡fa - b \in \ker f. The same criterion holds for group homomorphisms and for linear maps.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionKernel of a ring homomorphism, definition, no evidence levelKernel of a ring homo…definitionA ring homomorphism is injective iff its kernel is zero, statement, no evidence levelA ring homomorphism i…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. Ring homomorphism definition
  4. Kernel of a ring homomorphism definition
  5. A ring homomorphism is injective iff its kernel is zero statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ring_homomorphism, Properties
About this page
Tag0MMCB8RXGA
URIm2://0MMCB8RXGA
Kindstatement, proposition
WritingW0 stub
MSC16B99
StatusThis copy of the site has no build data. Run m2 status 0MMCB8RXGA.
UsesRing homomorphism, Kernel of a ring homomorphism

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