In [2, Section 1.6] Peskine and Szpiro prove a theorem on adic approximations of finite free resolutions over local rings which, together with M. Artin's Approximation Theorem [1], allows them to "descend" modules of finite projective dimension over the completions of certain local rings to modules of finite projective dimension over finite etale extensions of those rings. In this note we will ...

Suppose M is a maximal ideal of a commutative integral domain R and that some power Mn of M is finitely generated. We show that M is finitely generated in each of the following cases: (i) M is of height one, (ii) R is integrally closed and htM = 2, (iii) R = K[X; S̃] is a monoid domain over a field K, where S̃ = S ∪ {0} is a cancellative torsion-free monoid such that ⋂∞ m=1 mS = ∅, and M is the m...

Let R be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of R by means of local j-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion A ⊆ B of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multipl...

We prove that the zero-set of a C∞ function belonging to a noetherian differential ring M can be written as a finite union of C∞ manifolds which are definable by functions from the same ring. These manifolds can be taken to be connected under the additional assumption that every zero-dimensional regular zero-set of functions in M consists of finitely many points. These results hold not only for...

We study skew inverse power series extensions R[[y−1; τ, δ]], where R is a noetherian ring equipped with an automorphism τ and a τ -derivation δ. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete, local, ...

We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R. Recall that a prime ideal P of R is σ-divided if it is comparable (under inclusion) to every σ-stable ideal I of R. A ring R is called a σ-divided ring if every prime ideal of R is σ-divided. Also a ring R is almost σ-divided r...

Zaks (1969) proved that the answer is affirmative for a left and right noetherian ring if both dimensions are finite. Such rings are called Gorenstein. For a positive integer k, Auslander and Reiten (1994) initiated the study of k-Gorenstein algebras, which has stimulated several investigations. They showed that the answer to the question above is positive in case is an artin -Gorenstein algebr...

We prove various extensions of the Local Flatness Criterion over a Noetherian local ring R with residue field k. For instance, if Ω is a complete R-module of finite projective dimension, then Ω is flat if and only if Torn (Ω, k) = 0 for all n = 1, . . . , depth(R). In low dimensions, we have the following criteria. If R is onedimensional and reduced, then Ω is flat if and only if Tor1 (Ω, k) = ...

3 Topics in Commutative Algebra 2 3.1 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Quotient Rings and Homomorphisms . . . . . . . . . . . . . . 5 3.4 The Characteristic of a Ring . . . . . . . . . . . . . . . . . . . 7 3.5 Polynomial Rings in Several Variables . . . . . . . . . . . . . . 7 3.6...

Let R be a Noetherian ring . We prove that has global dimension at most two if, and only every prime ideal of is linear type. Similarly, we show three syzygetic. As consequence, derive characterization these rings using the André-Quillen homology.