Let R be a commutative Noetherian ring, a an ideal of R, M and N be two finitely generated R-modules. Let t be a positive integer. We prove that if R is local with maximal ideal m and M ⊗R N is of finite length then H t m (M, N) is of finite length for all t ≥ 0 and lR(H t m (M, N)) ≤ ∑t i=0 lR(Ext i R (M, H m (N))). This yields, lR(H t m (M, N)) = lR(Ext t R(M, N)). Additionally, we show that ...

<p>For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on C ∗ (X, its subring bounded functions, and Σ(X, denote collection all rings that lie between C). We show there is natural correlation absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in P(X, their real-valued counterparts ∩ C(X). These corr...

We show that cores of ideals do not preserve the inclusion. Huneke and Swanson[2] raised the question of whether, given integrally closed ideals I ⊂ I ′ in a ring R, it is necessarily true that core(I) ⊂ core(I ). Hyry and Smith [4, Corollary 5.5.1] gave a partial answer. The purpose of this note is to show that the answer is no in general. Although the simplest counterexample is a principal id...

There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Gorenstein. A more general invariant of an algebra or finite A-module M we consider here is the set Jordan types elements maximal ideal

In this article‎, ‎we have characterized ideals in $C(X)$ in which‎ ‎every ideal is also an ideal (a $z$-ideal) of $C(X)$‎. ‎Motivated by‎ ‎this characterization‎, ‎we observe that $C_infty(X)$ is a regular‎ ‎ring if and only if every open locally compact $sigma$-compact‎ ‎subset of $X$ is finite‎. ‎Concerning prime ideals‎, ‎it is shown that‎ ‎the sum of every two prime (semiprime) ideals of e...

Introduction Let B = k[x 1 ,. .. , x n ] be a polynomial ring over a field k and A = B/J a quotient ring of B by a homogeneous ideal J. Let m denote the maximal graded ideal of A. Then the Rees algebra R = A[mt] may be considered a standard graded k-algebra and has a presentation B[y 1 ,. In this paper we want to compare the ideals J and I J as well as their homological properties.

Let M be a module over Noetherian local ring A. We study M-independent sequences of elements the maximal ideal mA in sense Lech and Hanes. The main tool is new characterization M-independence sequence terms associated Koszul complex. As applications, we give result linkage theory, freeness criterion for existence strongly length equal to embedding dimension edim(A) A, another inspired from patc...