## Properties of Quotients

"1. Homomorphiesatz, Projektionssatz" (Universal mapping property)
Let f:A→B be a homomorphism of Σ-algebras
• π:A↠A/~f is a surjective (canonical) homomorphism.
• g̅:A/~f→B homomorph ⇔ g̅∘π homomorph (and g̅ surjective ⇔ g̅∘π surjective).
• ∃!f̅:A/~f→B homomorph such that f = f̅∘π.
• f̅ is injective.
• A/~f ≅ Im(f) (≅A/Ker(f) for groups).
Alternatively: (Universal mapping property of factor groups)
Let f:G→G' be a homomorphism of groups, with N⊴G and N⊆Ker(f)
• ∃!f̅:G/N→G' homomorph of groups such that f = f̅∘π
The same goes true for homomorphisms of rings with ideals N⊴G, and for R-linear mappings with sub-R-modules N≤G, etc. So in these cases TEns; G'↦{φ∈HomT(G,G') ¦ N⊆Ker(φ)} has G/N as presenting object.
"2. Homomorphiesatz, Einbettungssatz"
Let B⊆A be Σ-algebras, ~ ⊆ A×A a congruence relation, and B̅ := ⋃b∈B[b]~ = (π-1∘π)(B) ⊆ A the closure (or saturation) with respect to ~.
• B/~∩B×B ≅ B̅/~
"3. Homomorphiesatz, Serial Decomposition, refinement"
Let A be a Σ-algebra, and σ⊆ A×A a congruence.
• The congruences on A/σ are τ/σ for a congruence τ⊇σ on A.
• ∃!f:A/σ↠A/τ epimorphism such that πτ = f∘πσ
• ~f = τ/σ
• (A/σ)/(τ/σ) ≅ A/τ
"4. Homomorphiesatz, Parallel Decomposition ≙Chinese Remainder Theorem?"
Let A be a Σ-algebra, σ ⊆ A×A, τ ⊆ A×A  two independent congruences,
i.e. <σ,τ>EquivalenceClosure = 1 = ⊤ =  A×A.
• A/σ × A/τ ≅ A/σ∩τ
Chinese Remainder Theorem
Let R be a commutative ring with one, and I1,…,In⊴R be ideals. Then the following conditions are equivalent.
1. I1,…,In are coprime, i.e. ∀ν≠μ Iν+Iμ:=(Iν∪Iμ) = R
2. ⇔ R/ν=1,...,n Iν ≅ ∏ν=1,...,n R/Iν (injective by construction)
• ν=1,...,n Iν = I1⋅ … ⋅In (equivalent to (i) and (ii) if R is a principal ideal integrity domain)