Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 4
Search results for: S-nice subalgebras
4 Integral Domains and Alexandroff Topology
Authors: Shai Sarussi
Abstract:
Let S be an integral domain which is not a field, let F be its field of fractions, and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R ∩ F = S and F R = A. A topological space whose set of open sets is closed under arbitrary intersections is called an Alexandroff space. Inspired by the well-known Zariski-Riemann space and the Zariski topology on the set of prime ideals of a commutative ring, we define a topology on the set of all S-nice subalgebras of A. Consequently, we get an interplay between Algebra and topology, that gives us a better understanding of the S-nice subalgebras of A. It is shown that every irreducible subset of S-nice subalgebras of A has a supremum; and a characterization of the irreducible components is given, in terms of maximal S-nice subalgebras of A.Keywords: Alexandroff topology, integral domains, Zariski-Riemann space, S-nice subalgebras
Procedia PDF Downloads 1053 The Weights of Distinguished sl2-Subalgebras in Dn
Authors: Yassir I. Dinar
Abstract:
We computed the weights of the adjoint action of distinguished sl2-triples in Lie algebra of type Dn using mathematical induction.Keywords: lie algebra, root systems, representation theory, nilpotent orbits
Procedia PDF Downloads 2922 Algebras over an Integral Domain and Immediate Neighbors
Authors: Shai Sarussi
Abstract:
Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R∩F = S and the localization of R with respect to S \{0} is A. Denoting by W the set of all S-nice subalgebras of A, and defining a notion of open sets on W, one can view W as a T0-Alexandroff space. A characterization of the property of immediate neighbors in an Alexandroff topological space is given, in terms of closed and open subsets of appropriate subspaces. Moreover, two special subspaces of W are introduced, and a way in which their closed and open subsets induce W is presented.Keywords: integral domains, Alexandroff topology, immediate neighbors, valuation domains
Procedia PDF Downloads 1711 Integral Domains and Their Algebras: Topological Aspects
Authors: Shai Sarussi
Abstract:
Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R∩F = S and the localization of R with respect to S \{0} is A. Denoting by W the set of all S-nice subalgebras of A, and defining a notion of open sets on W, one can view W as a T0-Alexandroff space. Thus, the algebraic structure of W can be viewed from the point of view of topology. It is shown that every nonempty open subset of W has a maximal element in it, which is also a maximal element of W. Moreover, a supremum of an irreducible subset of W always exists. As a notable connection with valuation theory, one considers the case in which S is a valuation domain and A is an algebraic field extension of F; if S is indecomposed in A, then W is an irreducible topological space, and W contains a greatest element.Keywords: integral domains, Alexandroff topology, prime spectrum of a ring, valuation domains
Procedia PDF Downloads 123