Locally Compact Simple Rings having Minimal Left Ideals

In a recent paper2 we have called a ring 2 primitive if 2 contains a maximal right ideal a such that the quotient (a: W) = (0). The quotient (: 1) is defined to be the largest two-sided ideal of 21 that satisfies the condition 21(a: 21) C a. Thus if 21lias an identity, (a: 21) is the largest two-sided ideal of 21 contained in a. Primitive rings appear to play a role in the general structure theory of rings analogous to that played by simple rings in the Wedderburn-Artin theory. Primitive rings are also fundamental in general representation theory since it is known that a necessary and sufficient condition that a ring 21 be primitive is that 21 be isomorphic to an irreducible ring W of endomorphisms in a commutative group T. If 2 is irreducible in T and Zi is the division ring of endomorphisms commutative with the elements of A, then R is a dense ring of linear transformations in T regarded as a vector space over Z.3 We shall call 2 a left primitive ring if 2 contains a maximal left ideal a' such that the quotient (}': 21)l = (0). Here (s': 2) 1 is the largest two-sided ideal in 2 having the property (a': 21)1 21 C a'. Evidently, a ring is left primitive if and only if it is anti-isomorphic to an irreducible ring of endomorphisms. It is an open question whether or not a primitive ring is necessarily left primitive. This is indeed the case if 2 contains minimal ideals. In many other respects the theory of primitive rings that contain minimal ideals constitutes the most satisfactory part of the theory of primitive rings. An appropriate tool for studying rings of this type is a generalization of the duality theory previously used by Dieudonn6 in studying simple rings that possess minimal ideals.4 In this note we develop this generalization of Dieudonn6's results. We also investigate certain natural topologies that can be defined in any primitive ring.

In (1), Atkinson characterized the set of Fredholm operators on a Banach space X as those bounded operators invertible modulo the two-sided ideal of compact operators on X. It follows from this characterization that the Fredholm operators can also be described as those bounded operators which are invertible modulo the two-sided ideal of bounded operators on X which have finite-dimensional range. This ideal is the socle of the algebra of all bounded operators on X. Now, if A is any ring with no nilpotent left or right ideals, then the concept of socle makes sense (the socle of A in this case is the algebraic sum of the minimal left ideals of A, or 0 if A has no minimal left ideals). Also, in this case, the socle is a two-sided ideal of A. In this paper we study the elements in a ring A which are invertible modulo the socle. We call these elements the Fredholm elements of A.

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 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.

In [10], it is defined that a right (or left) ideal I of a ring R is very large if the cardinality of R/I is finite. It is also proven in [10, Theorem 3.4] that if R is a prime ring with 1 such that its characteristic is zero, then R is a right order in a simple ring with the minimum condition on one sided ideals if every large right ideal of R is very large. In the present note, we shall prove that if R is a prime ring with 1 such that its characteristic is zero and R is also a compact topological ring, then R is a right and left order in a simple ring with the minimum condition on one sided ideals, which is also a non-discrete locally compact topological ring if and only if every large right ideal of R is open. In particular, if R is an integral domain with 1 (not necessarily commutative) such that its characteristic is zero, then R is openly embeddable [13, p. 58] in a locally compact (topological) division ring if and only if every large right ideal of R is open. Following S. Warner [13], we shall say R is openly embeddable in a quotient ring Q(R) if there is a topology on Q(R) which is compatible with its structure, which induces on R its given topology and for which R is an open subset.

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

Let R be a primitive ring with involution *. Thus R may be considered as an irreducible ring of endomorphisms of an additive abelian group V, so that D=HomR(V, V) is a division ring. Let C be the center of D. We shall furthermore assume that CRCR. It can be shown that an involution -y--+is induced in C which has the property that 1yx = (tyx*)* for all xER. The involution * is of the first kind if -y-+-7 is the identity mapping and is of the second kind if there is a -y Oe C such that fy = -y. The set of symmetric elements of R will be denoted by S. We now assume that S satisfies a nontrivial generalized polynomial identity over C (in the sense of Amitsur). This means that there exists a nonzero element f(xl, x2, * * *, x.) in the so-called C-universal product R(x) of the C-algebra R and the free C-algebra C[X,, x2,* * *, x,, ... ] in noncommuting indeterminants xi, X2, *. * Xn, ...* such thatf(s1, S2, * * *, sn) = 0 for all SI S2, *.*.*, SnCS. For more precise details concerning the above notions we refer the reader to [1, ?4], and [2, ?3]. The usual linearization process may be used so that we may assume without loss of generality that S satisfies a nontrivial generalized (homogeneous) multilinear identity of degree n in xi, x2, ,n:

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.

Structural Properties of Models of N1-Categorical Theories

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Let R R be a semiprime K K -algebra with unity, d d a nonzero derivation of R R , and f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) a monic multilinear polynomial over K K such that d ( f ( a 1 , … , a t ) ) ≠ 0 d(f({a_1}, \ldots ,{a_t})) \ne 0 for some a 1 , … , a t ∈ R {a_1}, \ldots ,{a_t} \in R . It is shown that if for every r 1 , … , r t {r_1}, \ldots ,{r_t} in R R either d ( f ( r 1 , … , r t ) ) = 0 d(f({r_1}, \ldots ,{r_t})) = 0 or d ( f ( r 1 , … , r t ) ) d(f({r_1}, \ldots ,{r_t})) is invertible in R R , then R R is either a division ring D D or M 2 ( D ) {M_2}(D) , the ring of 2 × 2 2 \times 2 matrices over D D , unless f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) is a central polynomial for R R . Moreover, if R = M 2 ( D ) R = {M_2}(D) , where 2 R ≠ 0 2R \ne 0 and f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) is not a central polynomial for D D , then d d is an inner derivation of R R .

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

Twenty-five years ago McCoy published a characterization of commutative subdirectly irreducible rings. This result was generalized by Thierrin to duo rings with the word “field” which appeared in McCoy's theorem replaced by “division ring”. The purpose of this note is to give another generalization in which the words “division ring” will be replaced by “simple ring with 1 ”. The techniques resemble those of McCoy and Thierrin.