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This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to their double branched covers. We also show that the cohomology annihilator of a Gorenstein local ring is contained in the cohomology annihilator of its Henselization and in dimension one the cohomology annihilator of its completion. Finally, we investigate a relation between the cohomology annihilator of a Gorenstein ring and stable annihilators of its noncommutative resolutions. [ABSTRACT FROM AUTHOR]

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring R with residue field k , stable cohomology modules Ext ˆ R n (k , k) , defined for n ∈ Z , have been studied by Avramov and Veliche. Stable cohomology carries a structure of Z -graded k -algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology Ext R (k , k) and a shift of Hom k (Ext R (k , k) , k). We use this information to characterize the rings R for which stable cohomology is graded-commutative. Stable cohomology is connected through an exact sequence to bounded cohomology. We use this connection to understand the algebra structure of Ext ˆ R (k , k) by investigating the structure of bounded cohomology Ext ‾ R (k , k) as a graded Ext R (k , k) -bimodule. [ABSTRACT FROM AUTHOR]

Abstract: This paper answers the question of Anderson, Frazier, Lauve, and Livingston: for which finite commutative rings R is the zero-divisor graph planar? We build upon and extend work of Akbari, Maimani, and Yassemi, who proved that if R is any local ring with more than 32 elements, and R is not a field, then is not planar. They left open the question: “Is it true that, for any local ring R of cardinality 32, which is not a field, is not planar?” In this paper we answer this question in the affirmative. We prove that if R is any local ring with more than 27 elements, and R is not a field, then is not planar. Moreover, we determine all finite commutative local rings whose zero-divisor graph is planar. [Copyright &y& Elsevier]

Abstract: A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, “a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring”. Therefore, an interesting natural question of this sort is “whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?” The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. [Copyright &y& Elsevier]

Abstract: In the given paper we prove that every automorphism of a Chevalley group of type over a commutative local ring with 1/2 is standard, i.e., it is a composition of ring and inner automorphisms. [Copyright &y& Elsevier]

Abstract: In this paper, starting with a commutative ring R and a proper ideal , we construct and study a new ring denoted by R⋈I. In particular, we prove that if R is a CM local ring, then R⋈I is Gorenstein if and only if I is a canonical ideal of R and we apply this construction to algebroid curves. [Copyright &y& Elsevier]

Let (R,m) be a Noetherian local ring of prime characteristic p. In this paper, as an extension of the concept of pn-basis, we introduce the notion of m-adic pn-basis, and we show that R/Rpn has an m-adic pn-basis for every n (n=1,2,…) if and only if R is a regular local ring. For any Noetherian local ring, further we introduce the concept of m-adic free graded algebra and we give a regularity criterion in terms of the higher differential algebra, and others. [Copyright &y& Elsevier]

Let R be a Cohen–Macaulay local ring with maximal ideal m. In this paper we present a procedure for computing the Ratliff–Rush closure of an m-primary ideal I⊂R. [Copyright &y& Elsevier]

In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring R is a complete intersection if and only if every complex of R -modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring R is locally a complete intersection if and only if every complex of R -modules with finitely generated homology is virtually small. [ABSTRACT FROM AUTHOR]