noetherian rings
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2021 ◽  
Vol 8 (28) ◽  
pp. 885-898
Author(s):  
Michael Loper

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.


Author(s):  
Jawad Abuhlail ◽  
Rangga Ganzar Noegraha

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no nonzero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so-called [Formula: see text]-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the [Formula: see text]-injective semimodule, and the [Formula: see text]-injective semimodules through several implications, examples and counter examples. Moreover, we show that every semimodule over an arbitrary semiring can be embedded in a [Formula: see text]-[Formula: see text]-injective semimodule.


Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
J. R. García Rozas ◽  
Luis Oyonarte

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.


Author(s):  
Hagen Knaf

A theorem of Lichtenbaum states, that every proper, regular curve [Formula: see text] over a discrete valuation domain [Formula: see text] is projective. This theorem is generalized to the case of an arbitrary valuation domain [Formula: see text] using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring [Formula: see text] of a point [Formula: see text] is called regular, if every finitely generated ideal [Formula: see text] has finite projective dimension. The generalization is a particular case of a projectivity criterion for proper, normal [Formula: see text]-curves: such a curve [Formula: see text] is projective if for every irreducible component [Formula: see text] of its closed fiber [Formula: see text] there exists a closed point [Formula: see text] of the generic fiber of [Formula: see text] such that the Zariski closure [Formula: see text] meets [Formula: see text] and meets [Formula: see text] in regular points only.


2021 ◽  
Vol 29 (2) ◽  
pp. 173-186
Author(s):  
Fuad Ali Ahmed Almahdi ◽  
El Mehdi Bouba ◽  
Mohammed Tamekkante

Abstract Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.


Author(s):  
Esmaeil Rostami

In this paper, we introduce a class of commutative rings which is a generalization of ZD-rings and rings with Noetherian spectrum. A ring [Formula: see text] is called strongly[Formula: see text]-Noetherian whenever the ring [Formula: see text] is [Formula: see text]-Noetherian for every non-nilpotent [Formula: see text]. We give some characterizations for strongly [Formula: see text]-Noetherian rings and, among the other results, we show that if [Formula: see text] is strongly [Formula: see text]-Noetherian, then [Formula: see text] has Noetherian spectrum, which is a generalization of Theorem 2 in Gilmer and Heinzer [The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980) 13–16].


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