In this note we show that a Noetherian module has a dual module if and only if it satisfies AB5*. A connection between completeness and AB5* is also established. In this note we relate completeness, quasi-completeness, the A B5* condition, and duality. The main result is that a Noetherian R-module has a dual module if and only if it satisfies A B5*. Throughout this note R will denote a commutat...

In his article: “Untersuchungen über die Teilbarkeitseigenschaften in Körpern” J. Reine Angew. Math. 168, 1 36, 1932 [21], Heinz Prüfer introduced a new class of integral domains, namely those domains R in which all finitely generated ideals are invertible. He also proved that to verify this condition, it suffices to check that it holds for all two-generated ideals of R. This was the modest beg...

For a coalgebraC , the rational functor Rat(−) : C∗ → C∗ is a left exact preradical whose associated linear topology is the family C , consisting of all closed and cofinite right ideals of C∗. It was proved by Radford (1973) that if C is right Noetherian (which means that every I ∈ C is finitely generated), then Rat(−) is a radical. We show that the converse follows if C1, the second term of th...

Let k be a field of positive characteristic, R an associative algebra over k and let ∆1,n = {δ1, . . . , δn} be a finite set of k-linear derivations from R to R. Let A = Rn = R[θ1, δ1] · · · [θn, δn] be an iterated differential operator k-algebra over R such that δj(θi) ∈ Ri−1θi +Ri−1; 1 ≤ i < j ≤ n. As central result we show that if R is noetherian affine ∆1,n-hypernormal and if Tauvel’s heigh...

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 ...

We study how Betti numbers of ideals in a local ring change under small perturbations. Given p ? N and given an ideal I Noetherian ( R , m ) our main result states that there exists > 0 such if J is with ? mod the same Hilbert function as then ? i / coincide for ? . Moreover, we present several cases which forced to have therefore numbers.

A result of Artin, Small, and Zhang is used to show that a noetherian algebra over a commutative, noetherian Jacobson ring will be Jacobson if the algebra possesses a locally finite, noetherian associated graded ring. This result is extended to show that if an algebra over a commutative noetherian ring has a locally finite, noetherian associated graded ring, then the intersection of the powers ...

Let k be an uncountable algebraically closed field and let A be a countably generated left Noetherian k-algebra. Then we show that A⊗k K is left Noetherian for any field extension K of k. We conclude that all subfields of the quotient division algebra of a countably generated left Noetherian domain over k are finitely generated extensions of k. We give examples which show that A⊗k K need not re...

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...