The GENERALIZED principal Ideal THEOREM is one of the cornerstones of the dimension theory for the Noetherian rings. For an R-module M, certain submodules of M that play a role analo- gous to that of prime ideals in the ring R are identied. Using this denition, we extend the this THEOREM to modules.

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\fa$, is non-zero. $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th GENERALIZED local cohomology module of $M$ and $N$ with respect to $\mathfrak a$, is zero for all $i$ with $i>p+c$. In this paper, we obtain the coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we obtain a necessary and sufficient condition for the vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the Lichtenbaum-Hartshorne vanishing THEOREM for GENERALIZED local cohomology modules.

In this paper, we introduce the notion of extended partial metric space and we present some fixed point THEOREMs in GENERALIZED partial metric spaces involving linear and nonlinear contractions.

Recently, Dhage [1] introduced the concept of D-metric. Afterwards, many authors [4, 5, 6] proved some fixed point THEOREMs in these spaces. In this paper, using the concept of D-metric, we define a Δ–distance on a complete D-metric space which is a generalization of the concept of ω–distance due to Kada, Suzuki and Takahashi [2]. This generalization is non trivial because a D-metric does not always define a topology, and even when it does, this topology is not necessarily Hausdorff (see [3] and [4, Ch.1]). Using the concept of Δ-distance, we prove a fixed point THEOREM, which is the main result of this paper.We state the definition of D–metric, Δ–distance and prove a lemma. For more information on D–metrics, we refer the reader to [1, 4, 6].