The Fredholm Elements of a Ring

The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).

Let A be an indiscrete locally compact primitive ring having minimal left ideals.Algebraically, A may be regarded as a dense ring of linear operators containing nonzero linear operators of finite rank on a vector space E over a division ring K, and furthermore if A is simple, then A contains only linear operators of finite rank.The topology of A induces topologies on F and A" in a natural way, so that K becomes a locally compact division ring and E a locally compact vector space over K. Theorems about locally compact division rings and locally compact vector spaces imply that if K is indiscrete, then E is necessarily finite dimensional over K, and thus A is the ring of all linear operators on a finite-dimensional vector space over a locally compact division ring.Kaplansky proved that if A has characteristic zero, then K is indeed indiscrete, but he showed by an example that E could be infinite dimensional.Kaplansky remarked [9, p. 459], however, that if A is simple, it seemed unlikely that F could be infinite dimensional, since otherwise "completeness appears to require the presence of linear transformations with infinitedimensional range."Our principal result in 1 is the verification of this conjecture; however, our argument is based not on the fact that a locally compact ring is complete, but rather on the fact that a locally compact space is a Baire space and on the symmetry between minimal left and right ideals in a simple ring.In 2 we shall present certain results about locally compact vector spaces over discrete division rings; these enable us to prove that a locally compact, indiscrete, central primitive algebra having minimal left ideals is finite dimensional provided its scalar field either is indiscrete, has characteristic zero, or is uncountably infinite.However, there is a very natural example of a locally compact, indiscrete, infinitedimensional central primitive algebra having minimal left ideals over a discrete countably infinite field of prime characteristic.1. Locally compact simple rings having minimal left ideals.We begin with a summary of the structure theory of a primitive ring A having minimal left ideals.Let e be a minimal idempotent of A, that is, an idempotent such that Ae is a minimal left ideal, or equivalently, such that eA is a minimal right ideal.Then eAe is a division ring, the additive group Ae becomes a right eAe-\tctor space by defining scalar multiplication to be the restriction of multiplication on A to

Let R be an associative ring with identity.Then the left socle of R is a direct summand of R as a right ϋNmodule if and only if it is projective as a left jR-module and contains no infinite sets of orthogonal idempotents.This implies, for example, that a ring with finitely generated left socle and no nilpotent minimal left ideals is a ring direct sum of a semisimple artinian ring and a ring with zero left socle.

The complete reduction of simple Clifford algebras is spanned by matrix bases. Considered as generators of an algebra the basis elements pij satisfy the product relations pijpkl= δjkpil. It follows that subsets of pij with subscript or superscript fixed span minimal left or right ideals. The expression of the basis elements pij in the usual multivector basis is derived from isomorphic matrix algebras. Spinors and Hermitian conjugate spinors are defined as members of minimal left and right ideals. Spinor scalar products are defined, these scalar products are in fact scalar multiples of an idempotent.

Let${\mathcal{A}}$be a semisimple Banach algebra with minimal left ideals and$\text{soc}({\mathcal{A}})$be the socle of${\mathcal{A}}$. We prove that if$\text{soc}({\mathcal{A}})$is an essential ideal of${\mathcal{A}}$, then every 2-local derivation on${\mathcal{A}}$is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and${\mathcal{J}}$-subspace lattice algebras, is a derivation.

In previous work, we associated to $$\text {SU(3)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>SU(3)</mml:mtext> </mml:math> , $$\textrm{G}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>G</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , and $$\text {Spin(7)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>Spin(7)</mml:mtext> </mml:math> -structures minimal left ideals for the Clifford algebras $$\mathbb {R}_{0,6},\mathbb {R}_{0,7}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , and $$\mathbb {R}_{0,8}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:msub> </mml:math> , respectively. In this paper, we continue to analyze the link between Berger’s classification theorem and the structure theorem of minimal left ideals for Clifford algebras of signature ( p , q ) by identifying $$\textrm{U}(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>U</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -structures with minimal left ideals for Clifford algebras of various signatures via the induced Kahler polynomial $$P(\omega _{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> associated with the symplectic form $$\omega _{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> that defines the $$\textrm{U}(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>U</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -structure as a stabilizer subgroup of $$\textrm{O}(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>O</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .

Algebras with minimal left ideals which are Hilbert spaces

Algebraic and topological equivalences in the Stone-Čech compactification of a discrete semigroup

In this article, we give a characterization of a class of bounded Fredholm operators on a Banach space which is developed to present some general existence results of the operators equation of the second kind. The obtained results are used to describe the Riesz–Schauder theory of compact operators in the more general setting of polynomially compact operators.

Let A be a commutative semi‐simple Banach algebra such that the set consisting of finite sums of elements from minimal left ideals coincides with that of finite sums of elements from minimal right ideals. Let S(A) (the socle of A) denote this set. Let C(A) denote the set of elements x in A such that the map a → xax is compact. It is shown that C(A) is the norm closure of S(A).

Flat endofinite modules, prime ideals, and duality

In this paper the essential spectra of closed, densely defined linear operators is characterized on a Banach spaces under perturbations of n-strictly power compact operators. Further we apply the obtained results to investigate the essential spectra of one-dimensional transport equation with general boundary conditions and the essential spectra of singular neutron transport equations in bounded geometries.

Given a homogeneous polynomial on a Banach space E belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of E and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron–Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allows us to obtain symmetric versions of some basic results of the metric theory of tensor products.

In this paper a structural similarity between a recent braid- and division algebraic description of the unbroken internal symmetries of a single generation of Standard Model (SM) fermions is identified. This unexpected connection between two independently motivated models provides the first step towards unifying them into a unified theory based on braid groups and normed division algebras (NDA).Each of the four NDAs over the reals is shown to contain a representation of a circular braid group. For the complex numbers and the quaternions, the represented circular braid groups are B2 and B3c, precisely those used to represent leptons and quarks as framed braids in the Helon model of Bilson-Thompson. It is then shown that the twist structure of these framed braids representing fermions coincides exactly with the states that span the minimal left ideals of the complex (chained) octonions, shown by Furey to describe one generation of leptons and quarks with unbroken SU(3)c and U(1)em symmetry.This identification of basis states of minimal ideals with certain framed braids is possible because the braiding in B2 and B3c in the Helon model are interchangeable. It is shown that the framed braids in the Helon model can be written as pure braid words in B3c with trivial braiding in B2, something which is not possible for framed braids in general.

The Cauchy problem for the first-order algebraic differential equation is considered A du/dt=(B+εC+ε^2 D)u(t,ε), u(t_0,ε)=u^0 (ε)∈E_1, where A,B,C,D are closed linear operators acting from a Banach space E_1 to a Banach space E_2 with domains everywhere dense in E_1, u^0 is a holomorphic function at the point ε=0, ε is a small parameter, t∈[t_0; t_max]. Such equations describe, in particular, the processes of filtration and moisture transfer, transverse vibrations of plates, vibrations in DNA molecules, phenomena in electromechanical systems, etc. The operator A is the Fredholm operator with zero index. The aim of the work is to study the boundary layer phenomenon caused by the presence of a small parameter. The necessary information and statements are given. A bifurcation equation is obtained. Two cases are considered: a) boundary layer functions of one type, b) boundary layer functions of two types. Newton’s diagram is used to solve the bifurcation equation. In both, the conditions under which boundary layer phenomenon arises are obtained — these are the conditions for the regularity of degeneracy. Case a) is illustrated by an example of the Cauchy problem with certain operator coefficients acting in the space R^4.