Let α be an endomorphism of an arbitrary ring R with identity. The aim of this paper is to introduce the notion of an α-rigid module which is an extension of the rigid property in rings and the α-reduced property in modules defined in [8]. The class of α-rigid modules is a new kind of modules which behave like rigid rings. A right R-module M is called α-rigid if maα(a) = 0 implies ma = 0 for any m ∈ M and a ∈ R. We investigate some properties of αrigid modules and among others we also prove that if M [x; α] is a reduced right R[x; α]-module, then M is an α-rigid right R-module. The ring R is α-rigid if and only if every flat right R-module is α-rigid. For a rigid right R-module M , M is α-semicommutative if and only if M [x; α] R[x; α] is semicommutative if and only if M [x; α] R[[x; α]] is semicommutative.

Let X be a C' fuzzy manifold and p be a point in X. At first, it is given that the tangent space at p denoted by 7'p(^) is a vector space. İn this paper, constructing the tangent bundle nx}= V T(X)on X, it is shown that there is a covariant functor from the category of C' fuzzy manifolds and fuzzy differentiable functions to the category of the tangent bundles on C' fuzzy manifolds and fuzzy manifold derivative functions.

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In this article, we would like to present a newly fuzzy contraction mapping and using it we would like to prove a fixed point theorem. In fact, we transfer this contraction mapping, first defined in metric spaces [15], and then transferred to fuzzy metric spaces [9] with modification, to extended fuzzy metric spaces [6]. And so we prove some fixed point theorems [9] existing in the literature in the new spaces [6].