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6.1. Properties of Schreier split epimorphisms of semirings 59
Thanks to Corollary 5.5.2, we have that the map q described in the corol- lary above is forced to be a morphism of monoids. However, it does not preserve the multiplication, in general, so it is not a morphism of semirings. The map q has the following properties:
Proposition 6.0.11. Given a Schreier split epimorphism (A,B,f,s) of semir- ings, we have:
(a) qk = 1K[f];
(b) qs = 0;
(c) q(0) = 0;
(d) if b ∈ B and x ∈ K[f], then q(s(b)·x) = s(b)·x and q(x·s(b)) = x·s(b); (e) for every a,a′ ∈A q(a·a′)=q(a)·q(a′)+sf(a)·q(a′)+q(a)·sf(a′).
Proof. The properties from (a) to (c) are immediate consequences of Lemma 6.0.8 and Proposition 2.1.5. Concerning (d), it suffices to observe that s(b) · x and x · s(b) belong to the kernel of f, so the thesis follows from (a). Finally, concerning (e), we have that q(a · a′) is the unique element of K[f] such that
a·a′ =q(a·a′)+sf(a·a′), so it suffices to prove that
q(a) · q(a′) + sf(a) · q(a′) + q(a) · sf(a′) + sf(a · a′) = a · a′. In fact we have:
q(a)·q(a′)+sf(a)·q(a′)+q(a)·sf(a′)+sf(a·a′) =
= q(a) · q(a′) + sf(a) · q(a′) + q(a) · sf(a′) + sf(a) · sf(a′) =
= (q(a) + sf(a)) · (q(a′) + sf(a′)) = a · a′.
6.1 Properties of Schreier split epimorphisms of semir- ings
All the following results, already proved for Schreier split epimorphisms of monoids, are still valid for semirings, thanks to Lemma 6.0.8.
Lemma 6.1.1. A Schreier split epimorphism of semirings is a strongly split epimorphism in the category SRng.