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 T→Ens;
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 ~.
⇒
-
"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.
⇒
- Chinese Remainder Theorem
- Let R be a commutative ring with one, and I1,…,In⊴R
be ideals. Then the following conditions are equivalent.
- I1,…,In are coprime, i.e. ∀ν≠μ
Iν+Iμ:=(Iν∪Iμ) = R
- ⇔ 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)