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...

Let (A,1) be a partially ordered set. The relation a 1 b can be read as “a precedes b”. For elements a and b in A, we write a ≺ b if and only if a 1 b and a 6= b holds. For notational convenience, we also write a o b if and only if a 1 b holds, and a  b if and only if a ≺ b holds. We call (A,1) a well-founded set if and only if every non-empty subset M of A contains at least one minimal elemen...

Meta-projective modules on a ring R were essentially studied by Feng Lianggui and Tong Wenting in [3] when R is a commutative ring. While Duo-rings were studied by A. L. Fall in [2] and by M. Sanghare in [8]. The results of [3] were extended by M. O. Abdelkader in [5] to the duo-ring case by providing characterizations of Duo-rings. In this note, we give an extension of the MP-dimension notion ...

In this paper we introduce techniques to gauge the torsion of the tensor product A ⊗R B of two finitely generated modules over a Noetherian ring R. The outlook is very different from the study of the rigidity of Tor carried out in the work of Auslander ([1]) and other authors. Here the emphasis in on the search for bounds for the torsion part of A⊗R B in terms of global invariants of A and of B...

In this paper we prove that ifM is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring (A,m), and if Him (M) = 0 except possibly for i ∈ {0, r, d} with some 0 ≤ r ≤ d, then there exists an integer l such that every parameter ideal for M contained in m has the same index of reducibility. This theorem generalizes earlier work of the second author...

A result of P. Samuel ([17] p. 186, Chap.II, Théorème 5) says that in a local noetherian ring (O,M) of Krull dimension d in which the residual field k is infinite, the multiplicity of a M-primary ideal I is equal to the multiplicity of an ideal (x1, . . . , xd) generated by some parameter sequence x1, . . . , xd contained in I. By a theorem of Rees ([16] p.142 Theorem 9.44), this implies that t...

Let (R,m) be a commutative Noetherian local ring. Suppose that M and N are finitely generated modules over R such that M has finite projective dimension and such that TorRi (M,N) = 0 for all i > 0. The main result of this note gives a condition on M which is necessary and sufficient for the tensor product of M and N to be a Cohen–Macaulay module over R, provided N is itself a Cohen–Macaulay mod...

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a σ-ample divisor, where σ is an automorphism of a projective scheme X. Many open questions regarding σ-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be σ-am...

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a σ-ample divisor, where σ is an automorphism of a projective scheme X. Many open questions regarding σ-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be σ-am...