Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Correspondence theorem: ideals of R/I correspond to ideals of R containing I

Statement

Correspondence theorem: ideals of R/I correspond to ideals of R containing I.

For I⊴R, J↦J/I and K↦π−1(K) are inverse inclusion-preserving bijections between ideals J of R containing I and ideals K of R/I.

Facts & Assumptions

Given: An ideal I⊴R and π:R→R/I.

[L1]

π is a surjective ring homomorphism with kernel I (The canonical projection R→R/I is a surjective ring homomorphism with kernel I).

[L2]

Kernels of ring homomorphisms are ideals (The kernel of a ring homomorphism is a two-sided ideal).

[L3]

Ideals are additive subgroups with absorption (Left, right and two-sided ideals).

[L4]

R/I is a ring (For a two-sided ideal I, the additive cosets form a ring R/I with identity 1+I).

Proof

technique · direct
1.1

If J contains I, then J/I=π[J] is an ideal of R/I; if K is an ideal of R/I, then π−1(K) is an ideal containing I.

L1L2L3L4givenconstruct
2.1

Surjectivity gives π[π−1(K)]=K, and I⊆J gives π−1(π[J])=J; inclusion is preserved.

step 1.1L1L2L3L4givenalgebra
3.1

Therefore the two assignments give the claimed correspondence.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

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Sources