It is proved that for a ring R either an affine algebra over field, or equicharacteristic complete local ring, some power of the Jacobian ideal annihilates ExtRd+1(−,−), where d Krull dimension R. Sufficient conditions are identified under which itself these Ext-modules, and examples provided show this not always case. The crucial new idea to consider derived version Noether different algebra.

Let (L, ≤), be an algebraic lattice. It is well-known that (L, ≤) with its topological structure is topologically scattered if and only if (L, ≤) is ordered scattered with respect to its algebraic structure. In this note we prove that, if L is a distributive algebraic lattice in which every element is the infimum of finitely many primes, then L has Krull-dimension if and only if L has derived d...

We study almost F-preenvelopes in the category of rings, for a significative class F of commutative rings. We completely identify those rings which have an almost F-preenvelope when F is the class of fields, semisimple rings, integer domains and local rings. We show that rings with Krull dimension zero have (almost) V-preenvelopes, where V is the class of von Neumann regular rings. Mathematics ...

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over...

Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen–Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. F. Carlson (Question 5.4 in [7]).

A (not necessarily commutative) Krull monoid—as introduced by Wauters—is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these re...