Central Idempotents Research Articles

On composition and absolute-valued algebras

In this note we give a complete description of the composition algebras A over fields of characteristic ≠ 2, 3 in the following cases: if A has an anisotropic norm and x2x = xxx2 for every element; when A has a unitary central idempotent, it satisfies the identity (x2x2)x2 = x2(x2x2), and A is of finite dimension or has anisotropic norm. As a consequence, we obtain the existence, up to an isomorphism, of only seven absolute-valued algebras with a non-zero central idempotent where the last identity holds. This result completes the study of the absolute-valued algebras of this kind that was initiated by El-Mallah and Agawany.We also introduce the class of e-quadratic algebra, which contains the quadratic algebras, but also includes large classes of composition and absolute-valued algebras. Many results on composition, absolute-valued and e-quadratic algebras are shown, and new proofs of some well-known theorems are given.

Power Centralized Semigroups

A semigroup is said to be power centralized if for every pair of elements x and y there exists a power of x commuting with y. The structure of power centralized groups and semigroups is investigated. In particular, we characterize 0-simple power centralized semigroups and describe subdirectly irreducible power centralized semigroups. Connections between periodic semigroups with central idempotents and periodic power commutative semigroups are discussed.

Möbius functions and semigroup representation theory

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon's approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role.

The State Space of K 0 of Exchange Rings

ABSTRACT For a directly finite exchange ring R which satisfies general comparability, we construct all extreme points of the state space S(V(R),⟨ R⟩), where V(R) denotes the monoid of all isomorphic classes of finitely generated projective R-modules. From this, we further prove that S((K 0(R),[R])) is affinely homeomorphic to M 1 +(BS(R)), where BS(R) denotes the spectrum of the Boolean algebra B(R) of all central idempotents in R, and M + 1(BS(R)) the set of all probability measures on BS(R). These generalize the corresponding results on regular rings. Particularly, all of our results hold for exchange rings with all the idempotents central.

Wedderburn decomposition of finite group algebras

We show a method to effectively compute the Wedderburn decomposition and the primitive central idempotents of a semisimple finite group algebra of an abelian-by-supersolvable group G from certain pairs of subgroups of G.

Factor Rings and Splitting Ideals of Self-Injective, CS and Baer Rings#

ABSTRACT We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5). These results also generalize results of the author in Faith (1985), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring. The results are applied in Section 6 to extend theorems of Faith (1996) characterizing VNR rings, and, as the title of Faith (1996) suggests, extend the conjecture of Shamsuddin.

Absolute Valued Algebras Satisfying ( x 2, x 2, x 2) = 0#

Let A be an absolute valued algebra. We prove that if A satisfies the identity (x 2, x 2, x 2) = 0 for all x in A, and contains a central idempotent e, that is ex = xe for all x in A, then A is finite dimensional. This result enables us to prove that if A satisfies (x 2, x 2, x 2) = 0 and admits an involution then A is finite dimensional. To show that our assumptions on A are essential we recall that in El-Mallah [El-Mallah, M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51:39–49] it was shown that the existence of a central idempotent in A is not a sufficient condition for A to be finite dimensional; and the example given in El-Mallah [El-Mallah, M. L. (2003). Semi-algebraic absolute valued algebras with an involution. Comm. Algebra 31(7):3135–3141] shows that there exist infinite dimensional semi-algebraic absolute valued algebras satisfying the identity (x 2, x 2, x 2) = 0.

On Monomial Characters and Central Idempotents of Rational Group Algebras#

We give a method to obtain the primitive central idempotent of the rational group algebra ℚG over a finite group G associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group. We also show that for abelian-by-supersolvable groups this method takes a particularly easy form that can be used to compute the Wedderburn decomposition of ℚG.

On Rings of Matrices Over Division Rings

Let be a subring of Mn (D) for some division ring D satisfying the following three conditions: (i) there exists a division subring K of D such that for all a ∈ K and all ; (ii) for every , there exist ; and (iii) A = 0 if . It is shown that contains a maximal central idempotent C such that , and if E in in the center of are minimal and is a division ring and . As a corollary, we extend a result due to Omladič–Radjabalipour–Radjavi for K-algebras generated by unital semigroups of n × n matrices with entries in a field D and traces in a subfield K.

Equalizers and Flatness Properties of Acts, II

In Comm. Algebra 30 (3) (2002), 1475–1498, Bulman-Fleming and Kilp developed various notions of flatness of a right act AS over a monoid S that are based on the extent to which the functor AS$\otimes -$ preserves equalizers. In Semigroup Forum 65 (3) (2002), 428–449, Bulman-Fleming discussed in detail one of these notions, annihilator-flatness. The present paper is devoted to two more of these notions, weak equalizer-flatness and strong torsion-freeness. An act AS is called weakly equalizer-flat if the functor AS$\otimes -$ “almost” preserves equalizers of any two homomorphisms into the left act SS, and strongly torsion-free if this functor “almost” preserves equalizers of any two homomorphisms from SS into the Rees factor act S(S/Sc), where c is any right cancellable element of S. (The adverb “almost” signifies that the canonical morphism provided by the universal property of equalizers may be only a monomorphism rather than an isomorphism.) From the definitions it is clear that flatness implies weak equalizer-flatness, which in turn implies annihilator-flatness, and it was known already that both of these implications are strict. A monoid is called right absolutely weakly equalizer-flat if all of its right acts are weakly equalizer-flat. In this paper we prove a result which implies that right PP monoids with central idempotents are absolutely weakly equalizer-flat. We also show that for a relatively large class of commutative monoids, right absolute equalizer-flatness and right absolute annihilator-flatness coincide. Finally, we provide examples showing that the implication between strong torsion-freeness and torsion-freeness is strict.

Representations of Association Schemes and Their Factor Schemes

In the present paper we investigate the relationship between the complex representations of an association scheme G and the complex representations of certain factor schemes of G. Our first result is that, similar to group representation theory, representations of factor schemes over normal closed subsets of G can be viewed as representations of G itself. We then give necessary and sufficient conditions for an irreducible character of G to be a character of a factor scheme of G. These characterizations involve the central primitive idempotents of the adjacency algebra of G and they are obtained with the help of the Frobenius reciprocity low which we prove for complex adjacency algebras.

An algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra

We present an algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra. This algorithm has been implemented for System GAP, version 4.

CENTRAL IDEMPOTENTS IN THE RATIONAL GROUP ALGEBRA OF A FINITE NILPOTENT GROUP

We describe the primitive central idempotents of a rational group algebra of a finite nilpotent group. The description does not make use of the character table of the group G.

The Terwilliger Algebra of the Hypercube

We give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube QDof dimension D. Let X denote the vertex set ofQD . Fix a vertex x∈X, and letT=T(x) denote the associated Terwilliger algebra. We show thatT is the subalgebra of MatX(C) generated by the adjacency matrixA and a diagonal matrix A*=A* (x), where A*has yy entryD− 2 ∂(x, y) for all y∈X , and where ∂ denotes the path-length distance function. We show that A andA* satisfy A2A*− 2AA*A+A*A2&=& 4A* , A*2A− 2A*AA*+AA*2&=& 4 A. Using the above equations, we find the irreducible T -modules. For each irreducible T -module W, we display two orthogonal bases, which we call the standard basis and the dual standard basis. We describe the action of A andA* on each of these bases. We give the transition matrix from the standard basis to the dual standard basis for W. We compute the multiplicity with which each irreducible T -module W appears inCX . We give an elementary proof that QDhas the Q -polynomial property. We show that T is a homomorphic image of the universal enveloping algebra of the Lie algebrasl2 (C). We obtain an element φ of T that generates the center ofT . We obtain the central primitive idempotents of T as polynomials in φ.

The Galois extensions induced by idempotents in a Galois algebra

Let B be a Galois algebra with Galois group G, Jg = {b ∈ B|bx = g(x)b for all x ∈ B} for each g ∈ G, eg the central idempotent such that BJg = Beg, and for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)( = {g ∈ G | g(eK) = eK}) containing K and the normalizer N(K) of K in G. An equivalence condition is also given for G(eK) = N(K), and BeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1 − eG).

Idempotents in the Endomorphism Ring of an Ideal in a p-Extension of a Complete, Discrete Valuation Field with Residue Field of Characteristic p as a Galois Module

In this paper, the question concerning the existence of nontrivial idempotents in the endomorphism ring of an ideal in a p-extension of a complete discrete valuation field with residue field of characteristic p > 2 as a Galois module is considered. The nonexistence of nontrivial central idempotents for a non-Abelian totally widely ramified extension is proved. Bibliography: 4 titles.

Lifting to non-integral idempotents

In the first section of this paper, we illustrate (for a finite group G and the field of fractions K of a suitable complete dvr R) how to produce all central idempotents of Z( KG) from knowledge of the images of Z( RG) modulo certain powers of J( R). In the second section, we outline another approach to lifting elements closely related to idempotents in more general rings, which may be of wider interest.

ON EXCHANGE RINGS WITH BOUNDED INDEX OF NILPOTENCE

This paper studies exchange rings R such that R/J(R) has bounded index of nilpotence. We give several characterizations of such rings. We prove that if a semiprimitive exchange ring R has index n, then for any maximal two-sided I of R, if R/I has length n, then there exists a central idempotent element e in R such that eRe is an n by n full matrix ring over some exchange ring with central idempotents, and the restriction π from eRe to R/I is surjective.

Generalized periodic and generalized Boolean rings

We prove that a generalized periodic, as well as a generalized Boolean, ring is either commutative or periodic. We also prove that a generalized Boolean ring with central idempotents must be nil or commutative. We further consider conditions which imply the commutativity of a generalized periodic, or a generalized Boolean, ring.

Centralizers and Central Idempotents of Semigroup Rings

Let A be a nonempty subset of an associative ring R . Call the subring CR(A)={r∈ R\mid ra=ar \quadfor all\quad a∈ A} of R the centralizer of A in R . Let S be a semigroup. Then the subsemigroup S'= {s∈ S\mid sa=sb \quador\quad as=bs \quadimplies\quad a=b \quadfor all a,b∈ S} of S is called the C -subsemigroup. In this paper, the centralizer CR[S](R[M]) for the semigroup ring R[S] will be described, where M is any nonempty subset of S' . An non-zero idempotent e is called the central idempotent of R[S] if e lies in the center of R[S] . Assume that S\backslash S' is a commutative ideal of S and Annl(R)=0 . Then we show that the supporting subsemigroup of any central idempotent of R[S] must be finite.