Math 2.0

Associative rings and algebras Theorem

Second isomorphism theorem for rings

Builds on Quotient ringSubringRing isomorphism

Statement

Let RR be a ring, let SS be a subring of RR and let II be a two-sided ideal of RR. Then S+I={s+x:s∈S,x∈I}S + I = \{s + x : s \in S, x \in I\} is a subring of RR, and II is a two-sided ideal of S+IS + I. The intersection S∩IS \cap I is a two-sided ideal of SS. The map S/(S∩I)→(S+I)/IS/(S \cap I) \to (S + I)/I, s+(S∩I)↦s+Is + (S \cap I) \mapsto s + I, is a ring isomorphism.

Remarks

The proof applies the first isomorphism theorem to the map S→(S+I)/IS \to (S + I)/I, s↦s+Is \mapsto s + I. It is surjective, and its kernel is S∩IS \cap I. The theorem is also called the diamond theorem, after the picture of the four rings S∩IS \cap I, SS, II and S+IS + I.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionIdeal, definition, no evidence levelIdealdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionSubring, definition, no evidence levelSubringdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionQuotient ring, definition, no evidence levelQuotient ringdefinitionSecond isomorphism theorem for rings, statement, no evidence levelSecond isomorphism th…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. Subring definition
  6. Two-sided ideal definition
  7. Ring isomorphism definition
  8. Quotient ring definition
  9. Second isomorphism theorem for rings statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Isomorphism_theorems, Theorem B (rings)
About this page
Tag0HJ4DTW6FN
URIm2://0HJ4DTW6FN
Kindstatement, theorem
WritingW0 stub
MSC16D25
StatusThis copy of the site has no build data. Run m2 status 0HJ4DTW6FN.
UsesQuotient ring, Subring, Ring isomorphism

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