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Showing 1 to 17 of 17 for “"Commutative rings"”.
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Koszul Homology and Resolutions over Commutative Rings
<p>The Koszul homology algebra of a commutative local (or graded) ring R tends to reflect important information about the ring R and its properties. We examine Koszul homology and the relationship between Koszul homology and minimal free resolutions in various settings. We provide tools which allow …
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Zorn vector matrices over commutative rings and the loops arising from their construction
… finite fields can, in fact, be done over any commutative ring with the proper adjustments. The resulting loops are still Moufang, but no longer simple in general. Given a commutative ring and an ideal of that ring, the loop constructed over that ring can be decomposed into two pieces. In this …
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Categories of spaces built from local models
… For concreteness, consider the passage from commutative rings to schemes. There are three main steps: first, one identifies a distinguished class of ring homomorphisms corresponding to open immersions of schemes; second, one defines the notion of an open covering in terms of these …
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G-Frobenius manifolds
… To define a G-Frobenius manifold as a braided-commutative generalization of the ordinary commutative Frobenius manifold, we develop the theory of G-braided spaces. These are defined as G-graded G-modules with certain braided-commutative "rings of functions", generalizing the commutative rings …
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Derived algebraic geometry
… derived algebraic geometry based upon simplicial commutative rings. We define derived versions of schemes, algebraic spaces, and algebraic stacks. Our main result is a derived analogue of Artin's representability theorem, which provides a precise criteria for the representability of a moduli …
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Characterizations Of Zero Divisor Graphs Determined By Equivalence Classes Of Zero Divisors
We study zero divisor graphs of commutative rings determined by equivalence classes of zero divisors, specifically for a Noetherian ring R. We study the classification of these graphs. Specifically, we add more criteria to the list of characterizations that disqualify a graph as the zero divisor …
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Extensive categories, commutative semirings and Galois theory
We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, …
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Derived algebraic geometry over En̳-rings
We develop a theory of less commutative algebraic geometry where the role of commutative rings is assumed by En-rings, that is, rings with multiplication parametrized by configuration spaces of points in Rn. As n increases, these theories converge to the derived algebraic geometry of Tobn-Vezzosi …
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Representation of non-commutative topological algebras
… theorem enables us to represent a complex commutative C*-algebra as a full algebra of complex valued functions defined on its set of primitive ideals which is called the structure space of the algebra. In is thesis we are concerned with the generalization of this type of representation …
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SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO
… to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving …
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SIMPLE TWO-SIDED RATIONAL VECTOR SPACES OF RANK TWO
… to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving …
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Weyl Gröbner Basis Cryptosystems
… that is based on the theory of Gröbner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of Gröbner basis is, in general, EXPSPACE-hard. Cryptanalysis of these commutative Polly Cracker type cryptosystems is possible by using attacks …
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Closure Operations on Subgroups
… involving closure operations on the ideals of commutative rings. The most accessible paper on this is written by Neil Epstein, entitled "A Guide to Closure Operations in Commutative Algebra". This paper compiles much of the research done on the topic, and gives the reader an overview of closure …
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Isomorphism classes of almost completely decomposable groups
… In Chapter 4 we collect facts of matrices over commutative rings with an identity element. Matrices over the local ring (Z / p^e Z) of residue classes of the rational integers modulo a prime power play an important role. In Chapter 5 we introduce representing matrices of finite essential …
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Representation Theory of Orders over Cohen-Macaulay Rings
… <p>Orders are a certain class of noncommutative algebras over commutative rings. Originally</p> <p>defined by Auslander and Bridger, an R-order is an R-algebra which is a maximal CohenMacaulay</p> <p>R-module. In this thesis we consider orders, Λ, over Cohen-Macaulay local rings</p> …