Minimal Ideals Research Articles

Rings defined by $\mathcal{R}$-sets and a characterization of a class of semiperfect rings

Introduction. In this paper we give a characterization of semiperfect rings with projective, essential left socle (Theorems 5.1 and 4.4). This characterization is effected through the use of special a-sets. General a-sets are defined and discussed in ?3. Special a-sets are dealt with in ?4. Theorem 5.1 gives necessary and sufficient conditions for a ring with the above properties to be indecomposable, left (or right) perfect, semiprimary, left (or right) artinian, and for the left socle of the ring to be finitely generated. Thus we have, in particular, given a solution to a problem of Goldie [2, p. 268]: very interesting problem is the determination of artinian rings with zero singular ideal. In this connection, see also Gordon [4, Theorem 3.1]. Theorem 5.2 is a special case of Theorem 5.1-the determination of semiperfect rings with projective, essential left socle and a unique isomorphism class of minimal left ideals. The simplest natural instance of Theorem 5.2 is exploited in Theorem 5.3. Here we determine those semiperfect rings with projective, essential left socle in which principal indecomposable left ideals have unique simple submodules. This is a generalization of a theorem of Zaks' who handled the semiprimary case [9, Theorem 1.4, p. 67]. In Proposition 5.5 we give necessary and sufficient conditions for a semiperfect ring R with projective, essential left socle to have projective, essential right socle. This proposition implies that the right socle is typically not even a projective submodule of RR, Our main result in ?2 is the following lemma (Lemma 2.2): If R is a ring in which the identity is a sum of orthogonal idempotents ei, then the radical of R is left T-nilpotent if and only if the radical of eiRet is left T-nilpotent for every i. Also in ?2, we prove what might be a new lemma about reflexive, transitive relations on a finite set (see Lemma 2.7). This lemma allows us to give a normal form for rings of the type characterized in Theorem 5.3. An example (see Remarks 5.4) shows that this normal form simply need not occur in more general cases. We remark that this paper seems to inherently give rise to some apparently hard problems at various levels of abstraction. One glaringly obvious such instance:

The homotopic type of certain compact semigroups

The relation between homotopic types of compact linearly connected semigroups and their minimal ideals is investigated. In Theorem 1 we prove that, if a semigroup has a homotopic unit, even if it is only one-sided, then the minimal ideal is a deformation retract of the semigroup and so is homotopically equivalent to the semigroup. In Theorem 2 it is proved that a Q-semigroup is weakly homotopically equivalent to its minimal ideal.

Algebras with minimal left ideals which are Hilbert spaces

Algebras with minimal left ideals which are Hilbert spaces

Left ideal structure of C∗-algebras

An abelian C ∗-algebra is known to be isomorphic to the algebra of all complex continuous functions vanishing at infinity on its maximal ideal space. In an earlier paper the author defined a structure on the closed left ideals of an arbitrary C ∗-algebra which was analogous to the structure topology on the maximal ideals which exists for the abelian case. This paper carries that work forward. The first section expands the technical base of the theory by showing that more of the desirable properties of a compact Hausdorff space carry over to the new structure. The following section contains applications of various sorts. For example, Theorem I.1 gives a new characterization of AW ∗-algebras. The last section characterizes the maximal and minimal (closed) left ideals of any C ∗-algebra.

The socle of an alternative ring

The main facts about the minimal ideals and minimal right ideals of an associative ring are well known. In this paper we prove corresponding results for an alternative ring R. We make no restriction on the characteristic of R, but will often impose restrictions of semiprimeness type. (R is semiprime provided it has no ideal T such that T # (0) = P). Throughout this paper “ring” will mean “alternative ring”, and R will be a ring. We write A < R (A <, R; A <, R) to denote that A is an ideal (right ideal, left ideal) of R. If A is a minimal element of the set {M : (0) # M < R} (respectively, {M : (0) # M <,. R}), partially ordered by inclusion, we say that A is a minimal ideal (respectively, minimal fight ideal) of R, and write A <, R (respectively, A <,, R). Similarly for A <,, R. In Section 2, we show that if A <, R then either A2 = (0) or A is simple; this result is due in part to Zhevlakov. In Section 3 we similarly characterize A if A &. R. Roughly, if A2 # (0), then A is a minimal right ideal of the ideal C it generates in R, and, in general, C is simple. We also show that A is of the form eR, where e is a nuclear idempotent. In Section 4 we consider the right socle of R : S,(R) = Z(M : M <,, R). Under a suitable weak condition S,(R) is a two-sided ideal, and under a slightly stronger condition (weaker than semiprimeness of R) it coincides with the analogously defined left socle S,(R). In this situation we define the socle of R, S(R), to be S,(R) = S,(R). A structure theorem is proved for S,(R) which (like the results for minimal ideals and minimal right ideals) is particularly informative if R is purely alternative (free of nuclear ideals): that is, in some sense at the opposite extreme from being associative. We also show that S,(R) annihilates the Smiley radical M(R), thus justifying Baer’s name “antiradical” for S,(R). One result of Section 4 is that if A < R and R is semiprime, then S(A) = A n S(R). I n ec ion 5 we consider when a corresponding result S t

On quasimodular maximal left ideals of primitive rings generated by their minimal left ideals

On quasimodular maximal left ideals of primitive rings generated by their minimal left ideals

On the ideal structure of 𝐶(𝑋)

The ideal structure of C ( X ) C(X) , the algebra of continuous functions from a completely regular Hausdorff space X X to the scalars is analyzed by examining for fixed A ⊂ β X A \subset \beta X (the Stone-Čech compactification of X X ) the structure of the quotient I A / F A {I^A}/{F^A} , where I A [ F A ] {I^A}[{F^A}] is the ideal of maps f ∈ C ( X ) f \in C(X) for which \[ A ⊂ cl β X Z ( f ) [ A ⊂ int β X cl β X Z ( f ) ] . A \subset {\text {cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname {int} _{\beta X}}{\text {cl} _{\beta X}}Z(f)]. \] Unless it vanishes, I A / F A {I^A}/{F^A} has no minimal or maximal ideals, and its Krull dimension is infinite. If J J is an ideal of C ( X ) C(X) strictly between F A {F^A} and I A {I^A} , there are ideals J _ \underline {J} and J ¯ \bar J of C ( X ) C(X) for which F A ⊂ J _ ⊂ J ⊂ J ¯ ⊂ I A {F^A} \subset \underline {J} \subset J \subset \bar J \subset {I^A} with all inclusions proper. For K ⊂ C ( X ) K \subset C(X) , let Z ( K ) = ∩ { cl β X Z ( f ) : f ∈ K } Z(K) = \cap \{ {\text {cl} _{\beta X}}Z(f):f \in K\} . If J ⫋ I J \subsetneqq I are ideals of C ( X ) C(X) with Z ( J ) = Z ( I ) Z(J) = Z(I) and if I I is semiprime, there is an ideal K K strictly between J J and I I . If I I and J J are Z Z -ideals, K K can be chosen to be of the form P ∩ I , P P \cap I, P a prime ideal of C ( X ) C(X) . The maximal ideals of a semiprime ideal I I of C ( X ) C(X) are of the form I q ∩ I , q ∈ β X − Z ( I ) {I^q} \cap I,q \in \beta X - Z(I) . If A ⊂ β X A \subset \beta X is closed, I A {I^A} is a finitely generated ideal iff A A is open.

Rings in which minimal left ideals are projective

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.

On the equivalence of invariant integrals and minimal ideals in semigroups

Let S be a Hausdorff topological semigroup and Cb,(S), Cc (S), the spaces of real valued continuous functions on S which are respectively bounded and have compact support. A regular measure m on S is r*-invarient if m(B) = for every Borel B ⊂ S and every x ∈ S, where tx: s → sx is the right translation by x. The following theorem is proved: Let S be locally compact metric with the tx's closed. Then the following statements are equivalent: (i) S admits a right invariant integral on Cc (S). (ii) S admits an r*–invariant measure, (iii) S has a unique minimal left ideal. The above equivalence is considered also for normal semigroups and analogous results are obtained for finitely additive r*–invariant measures. Also in the case when S is a complete separable metric semigroup with the tx's closed, the following statements are equivalent: (i) S admits a right invariant integral I on Cb(S) such that I(1) = 1 and satisfying Daniel's condition. (ii) S admits an r*–invariant probability measure. (iii) S has a right ideal which is a compact group and which is contained in a unique minimal left ideal. Finally, in order that a locally compact S admit a right invariant measure, it suffices that S contain a right ideal F which is a left group such that (B ∩ F)x = BX ∩ Fx for all Borel B ⊂ S.

A generalization of quasi-Frobenius rings

As a generalization of quasi-Frobenius rings, G. Azumaya [2] has investigated a ring with the property that every faithful left module is a generator' in the category of left mnodules, and he has proved that such a ring is left self-injective and a direct sum of indecomposable left ideals having minimal left ideals. In his proof, however, the existence of the faithful injective module plays an important role, while the injective module is not necessarily finitely generated,2 even if the ring is a left Artinian ring. Therefore, there naturally arises a problem: Is an Artinian ring quasi-Frobenius if every finitely generated faithful module is a generator? The purpose of this paper is to give an affirmative answer to this problem as a direct consequence from a more general result which is rather similar to that of Azumaya stated above and related to perfect rings introduced by H. Bass [3]. Throughout this paper we shall assume that the ring R has identity element 1 and all modules over it are unital. For a subset A of R we shall denote

The Fredholm Elements of a 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.

Modular representations of split 𝐵𝑁 pairs

Introduction. In 1964 C. W. Curtis classified the absolutely irreducible modular representations of a large class of groups, the so called finite groups of Lie type. This was done by finding in each irreducible module an element, called a weight element, which is an eigenvector for certain elements in the modular group algebra; proving that two irreducible modules are isomorphic if and only if the corresponding eigenvalues (the collection of such being called a weight) are equal; and finally by determining which weights are associated with irreducible modules, i.e. which weights actually occur. This paper is a continuation of Curtis' work. We construct the absolutely irreducible modular representations of a finite group of Lie type by finding weight elements in the modular group algebra which generate a full set of nonisomorphic minimal left ideals. In some work of Steinberg and Curtis (see [14] and [6]) these representations were constructed for the covering groups of the Chevalley groups using the representations of the associated modular Lie algebras. The discussion given here avoids the Lie algebras altogether. Instead we construct each irreducible submodule of certain induced modules by finding the required weight elements. We prove at the same time that these induced modules have multiplicity free socles. Some remarks are made about the related questions of degrees and block structure of these representations. The finite groups of Lie type whose representations were classified by Curtis in [7] were defined by a large number of axioms. These apparently consisted of basic properties possessed by all known examples of such groups, the Chevalley groups and variations thereon as defined by Steinberg [13], for example. A simplified axiom scheme is used here, the heart of which is the axioms for groups with BN pairs (Tits [16]). The groups discussed in [7] satisfy the simplified axioms. The disadvantage of introducing new axioms is that the classification described in the first paragraph of this introduction must be redone. The proofs in this reworking are always similar to Curtis' but rarely identical, for the new axioms are enough weaker that some of the properties Curtis used do not follow from them. The advantages of reworking the classification theorem are twofold. First, the theorem is proved in greater generality. Secondly, the commutator relation is never assumed. It is usually replaced by arguments involving lengths of words in the Weyl group. The effect is to clarify the role of the root system in the classification theorem.

A note on the ideal structure of 𝐶(𝑋)

I is an ideal of C(X), and A = f {Z(f) =f-'(o) If EI}, then FACICIA. This means that for compact X, an examination of the ideal structure of C(X) can proceed piecewise: it is enough to obtain information about the ideals of the algebras IA/FA, A CX. For arbitrary X, such an approach will not tell all,' but it does serve to reveal the algebraic complexity of C(X). For we will show that unless it is trivial, IA/FA fails to contain minimal or countably generated maximal ideals, and fails to have minimum or maximum condition even on ideals of the form JIFA, J an ideal of C(X) lying between FA and IA. Corollary 2 is a partial generalization of [2, Theorem 5.2, p. 663] which we apply to show that the countably generated ideals of C(X) are rarely closed in the locally m-convex topology of uniform convergence on the compact parts of X (Corollary 3). LEMMA. If g&C(X)-FA, gE gIA+ FAPROOF. Suppose instead that g=gf+h for some f&IA, hE?FA. h vanishes on some neighborhood U containing A, and since V=f-'(B(O, 1/2))CU is a neighborhood of A and gGEFA, g(x)0o for some xCV. But thenf(x)=g(x)/g(x)=1, a contradiction. This means in particular that neither IA/FA nor IA has a (nonzero) identity. For choose gEI IA - FA. If either has an identity, there is an fEIA with fg-gEFA, and hence gEgIA+FA in violation of the lemma. So a distinction should be made between the ring and the algebra ideals of these algebras. Ideal will mean ring ideal. Of course if J is an ideal of C(X) lying between FA and IA, it is an algebra ideal of IA and J/FA is an algebra ideal of IA/FA. THEOREM 1. IA/FA has no (nonzero) minimal (ring or algebra) ideals. In fact if I is a nonzero ideal of IA/FA, there is an ideal J of C(X) lying between FA and IA such that O'J/FAcI.

On primitive $\Gamma $-rings with minimal one-sided ideals

On primitive $\Gamma $-rings with minimal one-sided ideals

On the Existence of Minimal Ideals in a Banach Algebra

On the Existence of Minimal Ideals in a Banach Algebra

On the existence of minimal ideals in a Banach algebra

htroduction. In this paper we give a sufficient condition that an algebra have a minimal left (or right) ideal. Specifically, we prove that if A is a complex semisimple Banach algebra with the property that the spectrum of every element in A is at most countable, then A has a minimal left ideal. If A is an A*-algebra, we prove that A has a minimal left ideal if the spectrum of every self-adjoint element of A is at most countable. These two basic results are given in ?2, and some applications of them are given in ??3 and 4. The main result of ?3 is Theorem 3.1 which is a variant of a theorem concerning B*-algebras that was proved through the separate efforts of M. A. Nalmark [5] and A. Rosenberg [7]. In ?4 we prove that a complex semisimple Banach algebra with the property that the spectrum of every element in the algebra has no nonzero limit points is a modular annihilator algebra. This together with a previous result of the author shows that modular annihilator Banach algebras are characterized by this property.

Rings faithfully represented on their left socle

In 1964 A. W. Goldie [1] posed the problem of determining all rings with identity and minimal condition on left ideals which are faithfully represented on the right side of their left socle. Goldie showed that such a ring which is indecomposable and in which the left and right principal indecomposable ideals have, respectively, unique left and unique right composition series is a complete blocked triangular matrix ring over a skewfield. The general problem suggested above is very difficult. We obtain results under certain natural restrictions which are much weaker than the restrictive assumptions made by Goldie. We characterize those rings in which the principal indecomposable left ideals each contain a unique minimal left ideal (Theorem (4.2)). It is sufficient to handle indecomposable rings (Lemma (1.4)). Such a ring is also a blocked triangular matrix ring. There exist r positive integers K 1 ,..., K r such that the i,j th block of a typical matrix is a K i x K j matrix with arbitrary entries in a subgroup D ij of the additive group of a fixed skewfield D. Each D ii is a sub-skewfield of D and D ri = D for all i. Conversely, every matrix ring which has this form is indecomposable, faithfully represented on the right side of its left socle, and possesses the property that every principal indecomposable left ideal contains a unique minimal left ideal. The principal indecomposable left ideals may have unique composition series even though the ring does not have minimal condition on right ideals. We characterize this situation by defining a partial ordering ρ on {i, 2,...,r} where we set iρj if D ij ≠ 0. Every principal indecomposable left ideal has a unique composition series if and only if the diagram of ρ is an inverted tree and every D ij is a one-dimensional left vector space over D ii (Theorem (5.4)). We show (Theorem (2.2)) that every ring A of the type we are studying is a unique subdirect sum of less complex rings A 1 ,...,A s of the same type. Namely, each A i has only one isomorphism class of minimal left ideals and the minimal left ideals of different A i are non-isomorphic as left A-modules. We give (Theorem (2.1)) necessary and sufficient conditions for a ring which is a subdirect sum of rings A i having these properties to be faithfully represented on the right side of its left socle. We show ((4.F), p. 42) that up to technical trivia the rings A i are matrix rings of the form [...]. Each Q j comes from the faithful irreducible matrix representation of a certain skewfield over a fixed skewfield D. The bottom row is filled in by arbitrary elements of D. In Part V we construct an interesting class of rings faithfully represented on their left socle from a given partial ordering on a finite set, given skewfields, and given additive groups. This class of rings contains the ones in which every principal indecomposable left ideal has a unique minimal left ideal. We identify the uniquely determined subdirect summands mentioned above in terms of the given partial ordering (Proposition (5.2)). We conjecture that this technique serves to construct all the rings which are a unique subdirect sum of rings each having the property that every principal-indecomposable left ideal contains a unique minimal left ideal.

Locally Compact Simple Rings having Minimal Left Ideals

Locally Compact Simple Rings having Minimal Left Ideals

Lorentz Equivalence, Unitary Symmetry, and Spin Unitary Symmetry

The enumeration and classification of the minimal ideals (spinor spaces) of the vector Clifford algebra generated from the vector basis of Lorentz space-time obtained previously is used in a discussion of equivalence mappings among these minimal ideals. Lorentz equivalence is defined as an equivalence mapping which possesses Lorentz symmetry. The concept of regular decomposition is introduced by which any element of the complete algegra is expressed in terms of a complete set of minimal ideals or spinor spaces. A regular current density which is invariant under unitary symmetry U4 among the spinor spaces of the regular decomposition is defined. The spin unitary symmetry, within a spinor space under which the spinor current density is invariant, is discussed.

Regular minimal sets. I

Abstract : A central problem in topological dynamics is the classification of minimal sets. In full generality, this problem appears to be quite difficult. Therefore, it seems appropriate to single out certain minimal sets which are 'well behaved' in some sense, and therefore more amenable to classification. It is for this purpose that we introduce the regular minimal sets in this paper. They are the universal minimal sets for certain ('admissible') properties. Several characterizations are given (theorem 3); the class of regular minimal sets is shown to coincide with the minimal right ideals of enveloping semigroups of transformation groups. and also with those minimal sets having a maximal supply of endomorphisms. (Author)