On Pullbacks of Abelian Groups and Pushouts of Rings

In this paper we generalize some results for pullbacks and pushouts known for C-algebras to the cases of Abelian groups or rings.


Introduction
Both pullbacks and pushouts are classical constructions in category theory.These constructions have been applied successfully also in other fields of mathematics, like Kasparov's KK-Theory or theory of vector bundles (where pullback is called the Whitney sum).
The present paper has its roots in the paper [3], where pullbacks and pushouts were used in order to study C * -algebras.While studying the results and proofs of [3], it appeared, that the proofs were carried out in quite a general manner and, therefore, did not depend so much on the topological structure.With slight modifications, it was possible to provide pure algebraic proofs without any topology considered.Even the need for algebras was superfluous -in fact, groups or rings for pushouts and Abelian groups for pullbacks were sufficient.
Since the results, presented in this paper, are of algebraic nature, then one can easily get similar results for more specific algebraic or topological structures as corollaries by replacing "Abelian groups" everywhere with "algebras" or "topological algebras" or by replacing "rings" with "Banach algebras" or "C * -algebras" etc.One must just demand that all homomorphisms involved here should be from the appropriate category, i.e., homomorphisms of algebras in case of algebras or homomorphisms of Banach algebras in case of Banach algebras, as well as kernels and cokernels exist.
For some reasons unknown to the author, most of the papers in category theory dealing with pullbacks and pushouts, do not consider the extensions or kernels so much.They tend to stay in the category of sets, which is not narrow enough to get the results presented in this paper.On the other hand, papers, which do have analogous results, are mostly written by people dealing with funtional analysis and considering only C * -algebras or Banach algebras, although the results hold in much wider case.The aim of the present paper was to give and to prove the results in as general form as possible.

Some results on pullbacks of Abelian groups
Let A, B and C be Abelian groups and maps α : We start with a generalization of Proposition 3.1 of [3, p. 256].As this result shows, it is enough to consider Abelian groups instead of C * -algebras.

Proposition 2.1. In a commutative diagram of Abelian groups X
X is a pullback of A and B along α and β if an only if the following conditions hold: Proof.Suppose that X with δ and γ in the commutative diagram above is a pullback of A and B along α and β .Then Since σ is an isomorphism, we get that x = θ X .Thus, (i) holds.
Suppose that the conditions (i), (ii) and (iii) hold and that the diagram above is commutative.Since A ⊕ C B is the canonical pullback of A and B along α and β , then, by the universal property of a pullback, there exists a homomorphism of Abelian groups σ, as defined above.Now, in order to show that X is a pullback of A and B along α and β , it is sufficient to show that σ is a bijection.
Take an element (a, b) , by (ii).Thus, there exists x ∈ X such that b = γ(x).Now, by using the properties of maps α, β , γ and δ, we get Therefore, a − δ(x) ∈ ker α = δ(ker γ), by (iii).Thus, there exists y ∈ X such that γ( y) = θ B and a − δ(x) = δ( y), whence a = δ(x + y).Using the linearity of δ and γ, we obtain Therefore, σ is onto, hence an isomorphism.Thus, X is a pullback of A and B along α and β .Remark 2.2.Similarly we can take instead of (ii) and (iii) the conditions (ii An extension 1 of Abelian groups (or rings) is a short exact sequence where A, B and C are Abelian groups (respectively, rings).Remember, that since this sequence is a short exact sequence, we get that α is injective, β is surjective and ker β = im α.
The proofs of Proposition 3.4 and Proposition 3.6 of [3] are written in such general way that they actually do not depend on the structure of C * -algebras.What one needs in these proofs is actually the fact, that the structures under consideration are Abelian groups.Therefore, we can just reformulate the results and suggest the reader to see the proofs of [3], where one can just replace the word "C * -algebra" with term "Abelian group" without any difficulty.Definition 2.3.We say that a class of Abelian groups satisfies property P if the following conditions are fulfilled: is closed under formation of extensions, i.e., if is a short exact of Abelian groups with A, B ∈ , then also X ∈ .

Lemma 2.4. Let be a class of Abelian groups which satisfies the property P. Then is closed under formation of pullbacks.
Again, following the proof of Proposition 3.6 of [3, pp.260-261], we obtain more general result 2 .

Proposition 2.5. Given two extensions of additive groups
where i = 1, 2, we obtain, by taking C = X /(A 1 + A 2 ), a third extension of additive groups

Some results on pushouts of rings
By a ring we will mean here an associative ring, which does not have to have a unital element.For the sake of correctness and in order to correct a misprint in the proof in [3], we mention, that the Noether Isomorphism Theorems for additive groups give us the isomorphisms between the additive groups Let A, B and C be rings and maps α : where X is a ring and maps δ : A → X and γ : B → X are homomorphisms of rings.If this diagram is commutative and for any ring Y and ring homomorphisms φ : A → Y , ψ : B → Y with φ • α = • β there exists unique map σ : X → Y with φ = σ • δ and ψ = σ • γ, then X (together with δ and γ) is called a pushout of A and B along α and β .Equivalently, in this case it is also said that the square above is a pushout square.
Last, we generalize Theorems 2.4 (part a) of Theorem 1) and 2.5 (part b) of Theorem 1) of [3], pp.248-249, from the case of C * -algebras to the case of rings.

Theorem 3.1. Consider a commutative diagram of extensions of rings
where ι A and ι C denote inclusions.Proof.(a) Since we have exact sequences in the diagram, we get that β is surjective.Moreover, ι A (A) = ker δ is a two-sided ideal of X and ι C (C) = ker β is a two-sided ideal of B because the kernel of a morphism of rings is always a twosided ideal.Take any x ∈ X .Then δ(x) ∈ E. Since β is surjective, then there exists Suppose that there exist ring Y and ring homomorphisms φ : A → Y and We have to show that there exists unique ring homomorphism σ : X → Y which fulfils the conditions Suppose that the exist a, a ′ ∈ A and b, b Then, using the linearity of ι A and γ, we get Using the linearity of φ and ψ and the fact obtained above, we see that ) ∈ X (in case we have algebras instead of rings, take also λ ∈ R).Then using again the linearity ι A and γ, we Similarly, using in addition the multiplicativity of φ and ψ, we obtain Next, using also the linearity of σ, we get Since it holds for any x ∈ X , we obtain = σ(x)σ( y) for any x, y ∈ X .
Hence, σ is multplicative, as well.Therefore, σ is a homomorphism of rings.Suppose, that there is another homomorphism of rings ω : Hence, ω = σ.Thus, we have shown that the left square is a pushout square.
, then we get also that ker δ = A ⊆ ker φ.
Suppose now, that the right square is a pushout square and denote A 0 = Id(α(C)) ⊆ A. Since A 0 is a two-sided ideal of X , then X /A 0 is a ring.Take Y = X /A 0 and consider the quotient where X is an Abelian group and maps γ : X → B and δ : X → A are homomorphisms of Abelian groups.If this diagram is commutative and for any Abelian group Y and Abelian group homomorphisms φ : Y → A, ψ : Y → B with α • φ = β • ψ there exists unique Abelian group homomorphism σ : Y → X with φ = δ • σ and ψ = γ • σ, then X (together with γ and δ) is called a pullback of A and B along α and β (for more general definition of a pullback, see [2], p. 71).It can be shown (see, for example, [1, p. 123], or [2, p. 72, Exercise 1]) that if X is a pullback of A and B along α and β , then X is isomorphic to a canonical pullback of A and B along α and β , which is defined as A ⊕ C B = {(a, b) ∈ A × B : α(a) = β (b)} and γ and δ are restrictions of the canonical projection maps.

1
Sometimes, the term "X is an extension of A along B" is used instead.For definitions of extensions of other types of algebraic structures, see, for example, [1, Definition 15.1.1,p. 121].

( a )
If A = α(C)A, i.e., α is a proper morphism, D = E and ε is the identity map, then the left square is a pushout square.(b) α(C) generates A as an ideal if and only if the right square is a pushout square.