Math 2.0

Associative rings and algebras Proof

Proof of the second isomorphism theorem for rings

Builds on RingSubringTwo-sided idealQuotient ringRing homomorphismKernel of a ring homomorphismFirst isomorphism theorem for ringsRing isomorphism

Proof

  1. Let RR be a ring, let SS be a subring of RR, and let II be a two-sided ideal of RR. Put T=S+I={s+x:s∈S,x∈I}T = S + I = \set{s + x : s \in S, x \in I}.
  2. The set TT contains 1=1+01 = 1 + 0, since 1∈S1 \in S and 0∈I0 \in I.
  3. Let s+xs + x and s′+x′s' + x' be in TT, with s,s′∈Ss, s' \in S and x,x′∈Ix, x' \in I. Then (s+x)−(s′+x′)=(s−s′)+(x−x′)(s + x) - (s' + x') = (s - s') + (x - x'), with s−s′∈Ss - s' \in S and x−x′∈Ix - x' \in I. So the difference lies in TT.
  4. The product is (s+x)(s′+x′)=ss′+(sx′+xs′+xx′)(s + x)(s' + x') = s s' + (s x' + x s' + x x'). Here ss′∈Ss s' \in S. Since II is a two-sided ideal of RR, the three terms sx′s x', xs′x s', xx′x x' lie in II. So the product lies in TT.
  5. By steps 2 to 4, TT is a subring of RR.
  6. Each x∈Ix \in I equals 0+x0 + x with 0=1−1∈S0 = 1 - 1 \in S. So I⊆TI \subseteq T. Also II is an additive subgroup, and for t∈Tt \in T and x∈Ix \in I, tx∈It x \in I and xt∈Ix t \in I, because II is a two-sided ideal of RR. So II is a two-sided ideal of TT.
  7. So the quotient ring T/IT/I exists. Define φ ⁣:S→T/I\varphi \colon S \to T/I by φ(s)=s+I\varphi(s) = s + I.
  8. The map φ\varphi is a ring homomorphism. By the definition of the operations of T/IT/I, φ(s+s′)=φ(s)+φ(s′)\varphi(s + s') = \varphi(s) + \varphi(s') and φ(ss′)=φ(s)φ(s′)\varphi(s s') = \varphi(s) \varphi(s'). Also φ(1)=1+I\varphi(1) = 1 + I, the identity of T/IT/I.
  9. For a,b∈Ta, b \in T, a+I=b+Ia + I = b + I if and only if a−b∈Ia - b \in I. Indeed, if the cosets are equal, then a∈b+Ia \in b + I. Conversely, if a−b=x∈Ia - b = x \in I, then a+y=b+(x+y)a + y = b + (x + y) and b+y=a+(y−x)b + y = a + (y - x) for each y∈Iy \in I.
  10. The map φ\varphi is surjective. Indeed, an element of T/IT/I is (s+x)+I(s + x) + I with s∈Ss \in S, x∈Ix \in I. Since (s+x)−s=x∈I(s + x) - s = x \in I, step 9 gives (s+x)+I=s+I=φ(s)(s + x) + I = s + I = \varphi(s).
  11. The kernel of φ\varphi is S∩IS \cap I. Indeed, by step 9, φ(s)=0+I\varphi(s) = 0 + I if and only if s∈Is \in I.
  12. By the first isomorphism theorem applied to φ\varphi, the kernel S∩IS \cap I is a two-sided ideal of SS.
  13. The same theorem, with step 10, shows that S/(S∩I)→T/IS/(S \cap I) \to T/I, s+(S∩I)↦φ(s)=s+Is + (S \cap I) \mapsto \varphi(s) = s + I, is a ring isomorphism.

Idea

Send each s∈Ss \in S to its coset modulo II in the bigger ring S+IS + I. Every coset of S+IS + I already has a representative in SS, so this map is onto. It kills exactly the elements of SS that lie in II, and the first isomorphism theorem does the rest.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first. The path takes the proofs Proof of the first isomorphism theorem for rings, Proof that a ring homomorphism is injective iff its kernel is zero, Proof of the universal property of the quotient ring.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionGroup, concept, no evidence levelGroupconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionRing, definition, no evidence levelRingdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIdeal, definition, no evidence levelIdealdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionSubring, definition, no evidence levelSubringdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionTwo-sided ideal, definition, no evidence levelTwo-sided idealdefinitionSubset, definition, no evidence levelSubsetdefinitionKernel of a ring homomorphism, definition, no evidence levelKernel of a ring homo…definitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionQuotient ring, definition, no evidence levelQuotient ringdefinitionProof of the universal property of the quotient ring, proof, no evidence levelProof of the universa…proofFunction, definition, no evidence levelFunctiondefinitionInjective function, definition, no evidence levelInjective functiondefinitionUniversal property of the quotient ring, statement, no evidence levelUniversal property of…statementProof that a ring homomorphism is injective iff its kernel is zero, proof, no evidence levelProof that a ring hom…proofA ring homomorphism is injective iff its kernel is zero, statement, no evidence levelA ring homomorphism i…statementProof of the first isomorphism theorem for rings, proof, no evidence levelProof of the first is…proofFirst isomorphism theorem for rings, statement, no evidence levelFirst isomorphism the…statementProof of the second isomorphism theorem for rings, proof, no evidence levelProof of the second i…proof
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. Element definition
  2. Group concept
  3. Proposition concept
  4. Set definition
  5. Ring definition
  6. Implication definition
  7. Predicate definition
  8. Ordered pair definition
  9. Ideal definition
  10. Ring homomorphism definition
  11. Subring definition
  12. Existential quantifier definition
  13. Universal quantifier definition
  14. Cartesian product definition
  15. Two-sided ideal definition
  16. Subset definition
  17. Kernel of a ring homomorphism definition
  18. Ring isomorphism definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Quotient ring definition
  22. Proof of the universal property of the quotient ring proof
  23. Function definition
  24. Injective function definition
  25. Universal property of the quotient ring statement
  26. Proof that a ring homomorphism is injective iff its kernel is zero proof
  27. A ring homomorphism is injective iff its kernel is zero statement
  28. Proof of the first isomorphism theorem for rings proof
  29. First isomorphism theorem for rings statement
  30. Proof of the second isomorphism theorem for rings proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Second_Isomorphism_Theorem_for_Rings
About this page
Tag1NM7SH936M
URIm2://1NM7SH936M
Kindproof
WritingW0 stub
MSC16D25
ProvesSecond isomorphism theorem for rings
UsesRing, Subring, Two-sided ideal, Quotient ring, Ring homomorphism, Kernel of a ring homomorphism, First isomorphism theorem for rings, Ring isomorphism
Changed4 November 2026

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