The aim of this note is to study the structure of a $\Gamma$-ring (not in the sense of Nobusawa) with minimum conditions. By ring theoretical techniques, we obtain various properties on the semi-prime $\Gamma$-ring and generalize Nobusawa's result which corresponds to the Wedderburn-Artin Theorem in ring theory. Using these results, we have that a $\Gamma$-ring with minimum right and left conditions is homomorphic onto the $\Gamma_{0}$-ring $\sum_{i=1}^{q}D_{n(i),m(i)}^{(i)}$, where $D_{n(i).m(i)}^{(i)}$ is the additive abelian group of the all rectangular matrices of type $n(i)\times m(i)$ over some division ring $D^{(i)}$, and $\Gamma_{0}$ is a subdirect sum of the $\Gamma_{i}$, $1\leqq i\leqq q$, which is a non-zero subgroup of $D_{m(i),n(i)}^{(i)}$ of type $m(i)\times n(i)$ over $D^{(i)}$.

In this paper, we prove that any locally trivial fibre bundles $p:X \to B$ with fibre $M$ a manifold modeled on an infinite-dimensional space $E$ (e.g. the Hilbert space $l_2$ or the Hilbert cube $Q$) is bundle isomorphic to the bundle $p * \mathrm{proj}:X \times E \to B$. Further, we can obtain a strong version of this Bundle Stability Theorem. From Bundle Stability Theorem, we can introduce the notion of deficiency in bundles. We show that a finite union of locally deficient sets is deficient and we prove a bundle version of Mapping Replacement Theorem.