Page 86 - Textos de Matemática Vol. 45

78 Chapter 6. Semirings
(v) x1 ·αl(b,x2)=αr(x1,b)·x2;
(vi) αr(αl(b1,x),b2)=αl(b1,αr(x,b2))
for any b,b1,b2 ∈ B and any x,x1,x2 ∈ X.
In the definition above, by bilinear we mean, as usual, that, for any ¯b ∈ B and any x¯ ∈ X, the restrictions of αl to {¯b}×X and B×{x¯} and of αr to {x¯} × B and X × {¯b} are linear. We have the following
Proposition 6.7.2. Given two semirings B and X, there is a natural bijection between the set of actions of B on X and the set of isomorphic classes of Schreier split epimorphisms with codomain B and kernel X.
Proof. We only give a sketch of the proof; we refer to [26] for more details. Given a Schreier split epimorphism
Xoo q //Aoo s //B kf
we define an action of B on X by putting
αl(b, x) = q(s(b) · x), αr(x, b) = q(x · s(b)).
Conversely, given an action α = (αl,αr), we define a semiring X α B, the semidirect product of B and X w.r.t. α, whose underlying set is the cartesian product X × B, the sum is componentwise and the multiplication is given by:
(x1,b1)·(x2,b2)=(x1 ·x2 +αl(b1,x2)+αr(x1,b2),b1 ·b2), and a Schreier split epimorphism
Xoo πX //X αBoo⟨0,1⟩ //B. ⟨1,0⟩ πB
When X is a trivial semiring, an action of B on X will be called a B- bimodule structure. Given two B-bimodules X and X′ (with actions α and α′, respectively), a semiring homomorphism h: X → X′ is a B-bimodule homo- morphism if
hαl (b, x) = αl′ (b, h(x)) and hαr (x, b) = αr′ (h(x), b) forallx∈X andb∈B.