We collect in this paper some remarks and observations about limit groups of equationally noetherian groups. We show in particular, that some known properties of limit groups of a free group or, more generally, of a torsion-free hyperbolic group can be seen as consequences of the fact that such groups are equationally noetherian. Especially, such properties are still true for linear groups and ...

Skolem and Nullstellensatz properties are analogues of the weak Nullstellensatz and Hilbert’s Nullstellensatz, respectively, for the ring of integervalued polynomials in several indeterminates Int(D) = {f ∈ K[x1, . . . , xn] | f(D) ⊆ D}, where D is a domain and K its quotient field. We show their equivalence when D is a Noetherian domain and extend the criterion of Brizolis and Chabert for Int(...

Suppose X is a locally noetherian Deligne–Mumford stack. Definition 1.2 has an obvious variant X̂ét using the underlying smaller étale site Xét and the restriction Oc Xét of Oc X to this site. By [3, 12.7.4], the category of cartesian Oc X -modules on Xlis-ét is equivalent to the category of Oc Xét-modules on Xét: (1.1) ModXlis-ét,cart(Oc X ) ' ModXét(Oc Xét) Definition 1.3. Let X be a locally n...

We organize this set of notes around a few theorems of Weierstrass. Write OCn for the sheaf of holomorphic functions on C. In the first three sections, we deduce the following “algebraic” results as consequences of the Weierstrass theorems: (i) for each open set Ω ⊂ C, the ring OCn(Ω) is not Noetherian, (ii) the local ring OCn,0 is factorial, (iii) the local ring OCn,0 is Noetherian. The two la...

We first provide an example of a ring R such that all possible 2 × 2 structural matrix rings over R are isomorphic. However, we prove that the underlying graphs of any two isomorphic structural matrix rings over a semiprime Noetherian ring are isomorphic, i.e. the underlying Boolean matrix B of a structural matrix ring M(B,R) over a semiprime Noetherian ring R can be recovered, contrary to the ...

The notion of intuitionistic fuzzy positive implicative hyper BCK-ideals of type-1, 2... 8 of hyper BCK-algebras was introduced in 2012 by Durga Prasad, Satyanarayana and Ramesh. In this paper, we investigate some related properties of intuitionistic fuzzy positive implicative hyper BCK-ideals of types-1. We characterize positive implicative Artinian (shortly, PI-Artinian) hyper BCK-algebras of...

This article consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this...

The theory of mixed multiplicities (not necessarily Noetherian) filtrations mR-primary ideals in a Noetherian local ring R, has been developed by Cutkosky, Sarkar and Srinivasan. objective this article is to generalise Minkowski-type inequality given their paper. We also recover result Srinivasan Verma as simple consequence our inequality.