Category of rings

terminology for this paper. In this paper x, y are sets, D is a non empty set, and U1 is a universal class. Let G, H be non empty double loop structures and let I1 be a map from G into H. We say that I1 is linear if and only if: (Def. 2) 1 For all scalars x, y of G holds I1(x+y) = I1(x)+I1(y) and for all scalars x, y of G holds I1(x · y) = I1(x) · I1(y) and I1(1G) = 1H. The following proposition is true (3) 2 Let G1, G2, G3 be non empty double loop structures, f be a map from G1 into G2, and g be a map from G2 into G3. If f is linear and g is linear, then g · f is linear. We consider ring morphisms structures as systems 〈 a dom-map, a cod-map, aFun 〉, where the dom-map and the cod-map are rings and the Fun is a map from the dom-map into the cod-map. In the sequel f is a ring morphisms structure. Let us consider f. The functor dom f yields a ring and is defined as follows: (Def. 3) dom f = the dom-map of f. The functor cod f yielding a ring is defined as follows: (Def. 4) cod f = the cod-map of f. The functor fun f yielding a map from the dom-map of f into the cod-map of f is defined as follows: (Def. 5) fun f = theFun of f. In the sequel G, H, G1, G2, G3, G4 are rings. Let G be a non empty double loop structure. One can check that idG is linear. Let I1 be a ring morphisms structure. We say that I1 is morphism of rings-like if and only if: 1 The definition (Def. 1) has been removed. 2 The propositions (1) and (2) have been removed