Yuichi Futa Shinshu University Nagano , Japan Hiroyuki Okazaki

We introduce Z-module structures which are extensions of additive loop structure and are systems 〈 a carrier, a zero, an addition, an external multiplication 〉, where the carrier is a set, the zero is an element of the carrier, the addition is a binary operation on the carrier, and the external multiplication is a function from Z× the carrier into the carrier. Let us mention that there exists a Z-module structure which is non empty. Let V be a Z-module structure. A vector of V is an element of V . In the sequel V denotes a non empty Z-module structure and v denotes a vector of V . Let us consider V , v and let a be an integer number. The functor a · v yields an element of V and is defined by: (Def. 1) a · v = (the external multiplication of V )(a, v).