Page 103 - Textos de Matemática Vol. 45

7.8. Baer sums in SRng 95
Let us now determine the direction of the special Schreier extension
A′ // k′ //X′ f′ //// Y.
The left action of y = f(x) on a′ = h(a) is, by definition, given by the term q(h¯(x)R[f′](h¯(x) · h¯(a))) = hqf (xR[f](x · a)), which is equal to f(x) · h(a), as we noticed above; but f(x) · h(a) = y · a′. The same holds for the right action. Accordingly the Y -bimodule structure on A′ is the desired one.
Suppose now that we have the morphism l of special Schreier extensions and the homomorphism m of Y -bimodules such that mh = K(l). Showing that l factors through h¯ is equivalent to showing that l coequalizes the equivalence relation S. So, suppose that we have xSz. Then xR[f]z and q(xR[f]z) ∈ K(h). Since f′′l = f, we get l(x)R[f′′]l(z), hence q′′(l(x)R[f′′]l(z)) + l(x) = l(z) by Lemma 3.1.4. Moreover
q′′(l(x)R[f′′]l(z)) = lq(xR[f]z) = mhq(xR[f]z) = m(1) = 1.
So we have l(x) = l(z), and hence a factorization μ: X′ → X′′ such that μh¯ = l,
so that K(μ)h = K(l) = mh. Since h is surjective, we get K(μ) = m. 7.8 Baer sums in SRng
Let Y be a semiring and A be a fixed Y -bimodule. We shall denote by SExt(Y,A) the set of isomorphic classes of special Schreier extensions with trivial kernel the underlying ring A and direction the Schreier split epimorphism (A Y, Y, pY , ιY ) associated with the given Y -bimodule structure on A. Similarly to what hap- pens in the category Mon of monoids we are going to show that SExt(Y,A) is endowed with a natural structure of abelian group.
For that, given a pair of classes of special Schreier extensions with trivial kernel A and same direction (A Y, Y, pY , ιY ):
A // kf //X f //// Y A // kf′ //X′ f′ //// Y
we can define the sum of the two previous classes as the class of the direct image of their product in T rSExtY (SRng) (given by their pullback above Y ) along the Y -bimodule homomorphism + : A × A → A given by the sum in the ring A:
kf×kf′ f×Y f′ A×A // //X×Y X′ //// Y
+ +˜
// // ′ ////
A X⊕YX ′Y kf′ f⊕Y f