Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥ 0, under the ring extension R⊂ R[X] (resp. R⊂ R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, in [11], holds in R[[X]], then it will holds in R.

Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal ideal m. The following conjecture proposed by Bass [1], has been proved by Peskin and Szpiro [2] for almost all rings: (B) If R admits a finitely generated R-module of finite injective dimension, then R is Cohen-Macaulay. The problems treated in this paper are closely related to the following generalization of Bass conjecture which is still wide open: (GB) If R admits a finitely generated R-module of finite Gorenstein-injective dimension, then R is Cohen-Macaulay. Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a finitely generated R-module of injective dimension ≤ 1. Levin and Vasconcelos indicate that if x m\m2 is a non-zerodivisor, then for every finitely generated R/xR-module M, there is id R M= id R/xr M+1. Using this fact, they construct a finitely generated R-module of finite injective dimension in the case where R is Cohen-Macaulay (the converse of Conjecture B). In this paper we study the Gorenstein injective dimension of local cohomology. We also show that if R is Cohen-Macaulay with minimal multiplicity, then every finitely generated module of finite Gorenstein injective dimension has finite injective dimension. We prove that a Cohen-Macaulay local ring has a finitely generated module of finite Gorenstein injective dimension.

In this article we will observe that Cm(X), the space C(X) with m - topology, is a regular Hausdorff space and we show that Cm(X) is first countable iff X is pseudo compact. We prove that Cm(X) ,is never a P - space (or pseudo compact). It is shown that the collection of zero divisor functions is closed in Cm(X) iff X is an almost P -space. The family of units in Cm(X) is investigated and using this, we observe that Cm(X) is never an extremely disconnected space. Finally, we show that C¥(X) is closed in Cm,(X) and the closure of Ck(X) is characterized in terms of maximal ideals of C(X).