Page 85 - Textos de Matemática Vol. 45

6.7. Semidirect products in SRng 77 see Section 4.4:
XXX
σ0k k ιX
p1 // φ¯
R×MSoo σ
OO 0
//S OO
//S(X) OO
p0
dS1 θX
R oo
ρX // M φ // M(X)
(dR0 p0,p) (p,dS1 p1)
dS0
dR1 //
dR0
All the commutative squares of the lower part are pullbacks, again see Section 4.4; accordingly both leftward whole lower rectangles are pullbacks. We are going to show that the two maps φdR0 and φdR1 are an index for the same split extension (R ×M S, R, p0, s0, σ0k) with kernel X. According to the property of the faithful split extensions, they will be equal, and consequently we shall get a factorization, which means R ⊂ R[φ]. So, let us denote by σ0 : S R ×M S the above horizontal map. Then necessarily σ0k is a kernel of p0 : R ×M S → R; so that we get p1σ0k = k and (p,dS1p1)σ0k = k since σ0 is a section of any map of the pair ((p, dS1 p1), p1). So, both φdR0 and φdR1 are an index of the split extension with kernel X in question.
6.7 Semidirect products in SRng
In analogy with Section 5.2, we now describe the explicit form of Schreier split epimorphisms of semirings by means of a semidirect product construction. In the case of semirings it is not possible to define an action of a semiring B on a semiring X simply as a morphism from B to a semiring depending only on X. We consider then the following
Definition 6.7.1. Given two semirings B and X, an action of B on X is a pair of bilinear maps (a left action and a right action)
αl : B × X → X and αr : X × B → X satisfying the following properties:
(i) αl(b,x1 ·x2)=αl(b,x1)·x2; (ii) αr(x1 ·x2,b)=x1 ·αr(x2,b);
(iii) αl(b1 · b2, x) = αl(b1, αl(b2, x)); (iv) αr(x,b1 ·b2)=αr(αr(x,b1),b2);