Properties of Quotients
 "1. Homomorphiesatz, Projektionssatz"
(Universal mapping property)

Let f:A→B be a homomorphism of Σalgebras
⇒
 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 Rlinear mappings with subRmodules N≤G, etc. So in these cases T→Ens;
G'↦{φ∈Hom_{T}(G,G') ¦ N⊆Ker(φ)} has G/N
as presenting object.

"2. Homomorphiesatz, Einbettungssatz"
 Let B⊆A be Σalgebras, ~ ⊆ A×A a congruence relation, and B̅ := ⋃_{b∈B}[b]_{~}
= (π^{1}∘π)(B) ⊆ A the closure (or saturation) with respect
to ~.
⇒

"3. Homomorphiesatz, Serial Decomposition, refinement"
 Let A be a Σalgebra, 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 π_{τ} = f∘π_{σ}
 ~_{f} = τ/σ
 (A/σ)/(τ/σ) ≅ A/τ

"4. Homomorphiesatz, Parallel Decomposition ≙Chinese Remainder
Theorem?"
 Let A be a Σalgebra, σ ⊆ A×A, τ ⊆ A×A two independent
congruences,
i.e. <σ,τ>_{EquivalenceClosure} = 1 = ⊤ = A×A.
⇒
 Chinese Remainder Theorem
 Let R be a commutative ring with one, and I_{1},…,I_{n}⊴R
be ideals. Then the following conditions are equivalent.
 I_{1},…,I_{n} are coprime, i.e. ∀ν≠μ
I_{ν}+I_{μ}:=(I_{ν}∪I_{μ}) = R
 ⇔ R/⋂_{ν=1,...,n} I_{ν} ≅ ∏_{ν=1,...,n}
R/I_{ν} (injective by construction)
⇒
 ⋂_{ν=1,...,n} I_{ν} = I_{1}⋅ … ⋅I_{n}
(equivalent to (i) and (ii) if R is a principal ideal integrity
domain)