Direct Sum Decomposition Research Articles

A subdirect decomposition of semiprime rings and its application to maximal quotient rings

Levy [2] has examined semiprime rings which are irredundant subdirect products of prime rings. In this note we look at the role of inessential prime ideals and see how every semiprime ring is a subdirect product of (i) a semiprime ring which is an irredundant subdirect product of prime rings, and (ii) a semiprime (nonprime) ring, all of whose prime ideals are essential. This leads to a direct sum decomposition of maximal left quotient rings of semiprime rings with left singular ideal zero.

Reducing Decompositions for Strictly Cyclic Operators

If $T$ is a strictly cyclic operator on $H$ then $H$ has a direct sum decomposition ${H_1} \oplus {H_2}$ where ${H_1}$ and ${H_2}$ are invariant under $T$ if and only if the spectrum of $T$ is not connected. If $\lambda$ is a reducing eigenvalue for the strictly cyclic operator $T$ then the multiplicity of $\lambda$ is one and $\lambda$ is an isolated point of the spectrum of $T$.

Algebraic L -Theory, I: Foundations

Introduction Where algebraic if-theory deals with modules, Ltheory considers modules with quadratic forms. The ^-groups are of interest to topologists because they are the surgery obstruction groups, as described by Wall ([6]). Although isomorphism groups of quadratic forms have been studied before, by Witt and others, the topological applications require new algebraic methods (cf. [5], [7]). The Z-groups Ln(7r) were obtained in [6] as the solutions to a specific topological problem, leaving open the question of the algebraic framework best suited for an '^-theory'. In [1] Novikov used algebraic if-theory and the formalism of hamiltonian physics to provide such machinery, though not as coherently as might be desired. In Part I of this paper we shall use the ideas of [1] to give the foundations of i-theory over a ring with involution, A. We shall define L-groups Un(A), Vn(A), and J^,(-4) as stable isomorphism groups of ' + forms' and '+ formations' involving finitely generated (f.g.) projective, stably f.g. free, and based A -modules respectively, depending on n (mod 4) only. Part II, [2], will be devoted to a detailed study of the L-groups of the Laurent extension ring Az = A[z,z ], with involution z K Z~. It will be shown that there exist natural direct sum decompositions

Topological classification of linear endomorphisms

Letf=f +⊕f −⊕f ∞⊕f 0 be a linear direct sum decomposition of a real endomorphismf∈ End ℝ n , where the components correspond respectively to absolute values of eigenvalues |λ|: 0<|λ|<1(f +); |λ|>1(f −); |λ|=0(f ∞); and |λ|=1(f 0). We conjecture that a complete set of invariants with respect to topological equivalence consists of the dimensions and orientations off + and off −, together with the linear isomorphy classes off ∞ andf 0. We prove that this is true in case ℝ n contains nof-periodic points of periods 5 or ≧7. The conjecture is also true in case it is true for all periodic rotations. In §9 we make some comments on this unsolved case. The main interest of the paper is in §6 and 7.

Topological classification of linear endomorphisms

Letf=f +⊕f −⊕f ∞⊕f 0 be a linear direct sum decomposition of a real endomorphismf∈ End ℝ n , where the components correspond respectively to absolute values of eigenvalues |λ|: 0<|λ|<1(f +); |λ|>1(f −); |λ|=0(f ∞); and |λ|=1(f 0). We conjecture that a complete set of invariants with respect to topological equivalence consists of the dimensions and orientations off + and off −, together with the linear isomorphy classes off ∞ andf 0. We prove that this is true in case ℝ n contains nof-periodic points of periods 5 or ≧7. The conjecture is also true in case it is true for all periodic rotations. In §9 we make some comments on this unsolved case. The main interest of the paper is in §6 and 7.

A class of pure subgroups of completely decomposable abelian groups

The class M of pure subgroups of completely decomposable torsion free abelian groups of finite rank is considered by Butler [2]. Many of the examples of pathological direct sum decompositions of finite rank torsion free abelian groups occur in the class .4. On the other hand, homogeneous a-groups are completely decomposable and if A is an s-group, then typeset(A) is finite and closed under intersection of types. This note considers a-groups with typesets of cardinality at most 4. Motivation is provided by the curious results of Cruddis [3]. An easier and more illuminating proof of these results is a consequence of the following theorems. Let A= {-To, TI, T2, T73} be a set of types with To =rl1r -72r lT3; let .A be the class of a-groups with typeset c A; let :A be the sum of types in A; and let >,A be the sum of types in typeset(A). Define T to be the type represented by (Xo) and, forp a prime, let T-D be the type represented by (ne) where n2,=O and nQ= xo if p#q.

A Decomposition for Combinatorial Geometries

A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte decomposition of a pregeometry into a subgeometry $G\backslash e$ and contraction $G/e$ is explored in a categorically integrated view using factored strong maps. After showing that direct sum decomposition distributes over the Tutte decomposition we construct a universal pair $(R,t)$ where $R$ is a free commutative ring with two generators corresponding to a loop and an isthmus; and $t$, the Tutte polynomial assigns a ring element to each pregeometry. Evaluations of $t(G)$ give the Möbius function, characteristic polynomial, Crapo invariant, and numbers of subsets, bases, spanning and independent sets of $G$ and its Whitney dual. For geometries a similar decomposition gives the same information as the chromatic polynomial throwing new light on the critical problem. A basis is found for all linear identities involving Tutte polynomial coefficients. In certain cases including Hartmanis partitions one can recover all the Whitney numbers of the associated geometric lattice $L(G)$ from $t(G)$ and conversely. Examples and counterexamples show that duals, minors, connected pregeometries, series-parallel networks, free geometries (on which many invariants achieve their upper bounds), and lower distributive pregeometries are all characterized by their polynomials. However, inequivalence, Whitney numbers, and representability are not always invariant. Applying the decomposition to chain groups we generalize the classical two-color theorem for graphs to show when a geometry can be imbedded in binary affine space. The decomposition proves useful also for graphical pregeometries and for unimodular (orientable) pregeometries in the counting of cycles and co-boundaries.

Exchange rings and decompositions of modules

In this paper, a new class of rings, called exchange rings, is defined and studied. The class of exchange rings includes regular rings in the sense of yon Neumann, local rings, and semiperfect rings. The chief property of these rings is that if S is any ring and M a left S-module which is a direct sum of finitely generated modules, each of which has an exchange ring as its endomorphism ring, then any two direct sum decompositions of M have isomorphic refinements. In particular, if R is an exchange ring, a projective R-module is a direct sum of ideals generated by idempotents. The main results of this paper are Theorems 2 and 3. Theorem 2 shows that a module has the finite exchange property (defined below) if and only if its endomorphism ring is an exchange ring. Theorem 3 gives a large class of rings which are exchange rings. The applications of Theorem 3 include generalizations of theorems proved by Kaplansky and Pierce for modules over regular rings. All rings will be associative rings with identity and all modules, unless otherwise specified, will be left modules. We will write the direct

A decomposition for combinatorial geometries

A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte decomposition of a pregeometry into a subgeometry G ∖ e G\backslash e and contraction G / e G/e is explored in a categorically integrated view using factored strong maps. After showing that direct sum decomposition distributes over the Tutte decomposition we construct a universal pair ( R , t ) (R,t) where R R is a free commutative ring with two generators corresponding to a loop and an isthmus; and t t , the Tutte polynomial assigns a ring element to each pregeometry. Evaluations of t ( G ) t(G) give the Möbius function, characteristic polynomial, Crapo invariant, and numbers of subsets, bases, spanning and independent sets of G G and its Whitney dual. For geometries a similar decomposition gives the same information as the chromatic polynomial throwing new light on the critical problem. A basis is found for all linear identities involving Tutte polynomial coefficients. In certain cases including Hartmanis partitions one can recover all the Whitney numbers of the associated geometric lattice L ( G ) L(G) from t ( G ) t(G) and conversely. Examples and counterexamples show that duals, minors, connected pregeometries, series-parallel networks, free geometries (on which many invariants achieve their upper bounds), and lower distributive pregeometries are all characterized by their polynomials. However, inequivalence, Whitney numbers, and representability are not always invariant. Applying the decomposition to chain groups we generalize the classical two-color theorem for graphs to show when a geometry can be imbedded in binary affine space. The decomposition proves useful also for graphical pregeometries and for unimodular (orientable) pregeometries in the counting of cycles and co-boundaries.

On Lebesgue-type decompositions for Banach algebras

If the maximal ideal space of a commutative complex unitary Banach algebra, X, is equipped with a nonnegative, finite, regular Borel measure, m, then for each element, x, in X, a. complex regular Borel measure, mx, can be obtained by integrating the Gelfand transform of x with respect to m over the Borel sets. This paper considers the possibility of direct sum decompositions of the form X = Ax ⊕ Px where Ax = {z ε X: mz ≪ mx} and Px = {z ε X: mz ┴ mx}.

Direct-sum decomposition of atomic and orthogonally complete rings

In this paper we give a necessary and sufficient condition for decomposition (as a direct sum of fields) of a ring R in which for every x ∈ R there exists a (and hence the smallest) natural number n(x) &gt; 1 such that . We would like to emphasize that in what follows R stands for a ring every element x of which satisfies (1).

Orthogonal decompositions of tensor spaces

1In this note we giv e an orthogonal direct sum decomposition of W in terms of the system of inequivalent irre ducible c haract ers of H. AMS S ubject Classifications: Primary, 1580; Secondary, 2080. Key wo rds: Irred ucibl e c haracte r ; sy mmetry class of tensors; sy mmetry operator; tensor product.

About the cohomology ring of a finite abelian group

Since the structure of finite abelian groups is so simple, one would expect that the cohomology ring H(G, R) of a finite abelian group G over a commutative ring R to be completely and explicitly known, particularly when the action of G on R is trivial and the ring R is some reasonable ring, say the ring Z of integers. Indeed considerable work has been accomplished in the computation of a homology of finite abelian groups in the context of Eilenberg-MacLane spaces by Eilenberg and MacLane [4]; a remarkable algebraic theory around this topic was built up by Cartan [2]. Further analysis of the homology ring of a finite abelian group is presented in [9]. However, as far as we can see, there still is no explicit and functorial description of H(G, R) in the general case. The following observations still do not fill this gap, but they contribute some new facets and perhaps offer a more direct approach to some results which are in the literature. Our approach is designed specifically to allow for easy generalization to the cohomology ring of a compact abelian group, which we discuss in this journal [o]. The full details and the proofs will appear elsewhere. Rather than to give too many technical details, we describe the essential features of our approach and try to point out where it differs from other methods. Each finite abelian group G decomposes into a direct sum of cyclic subgroups Gi® • • • @Gn of orders zit i = l, • • • , w, such that 2*|s*+i> 0<i<n; this standard decomposition is not unique, even though the Zi are. Since we want a description of H(G, R) which is functorial in G(and R), one is faced with two conflicting objectives: firstly, the final results must not depend on the given product decomposition; secondly, one practically has to use the structure theorem for G to obtain any reasonably explicit description for H(G, R). Thus one attempts to exploit a standard decomposition initially and then to eliminate the dependence of the direct sum decomposition by functorial methods. Thus our first step is to describe explicitly an

On a cohomology theory for pairs of groups

of crossed homomorphisms from G to A vanishing on H. Clearly this is a left-exact functor in the category GMJF of left G-modules. The nth rightderived functor of X(G, H, -) in Ga9Z isdenoted byHn+'(G, H, -). The group Hn(G, H, A), A C GM, is called the nth cohomology group of the pair (G, H) with coefficients in A. These groups were first described and studied by M. Auslander in [1], who also found the sequence of Proposition 1.2. In this note we prove an excision property for the functors Hn(G, H, -), Theorem 2.2, and we find a direct sum decomposition of them under suitable conditions, Propositions 2.3 and 2.5. From this one deduces by standard methods a Mayer-Vietoris type sequence for the cohomology of groups, Proposition 2.6. The results in this paper are part of the author's doctoral dissertation at the University of Rochester. The author wishes to thank C. E. Watts for his advise and encouragement.

On Class Groups of Free Products

If R is a commutative ring, let S(R) be the category of supplemented R algebras, where R is central in each object A of S(R). Thus, if SA: A t R is the augmentation, DA: R-NA the unit, so CAOCA = 1, then a morphism f: A-d F of S(R) is a ring homomorphism satisfying srf = S, f7be = 72r. We identify R with its image under I,, and denote the augmentation ideal Ker SA: Ah R by A. If A and F are objects of S(R), then their coproduct exists and is denoted A*rF. This is just the free product of A and F described in [3]. KJ(A) denotes the Grothendieck group of finitely generated projective left A modules, and K,(A) = KerSA *: KJ(A) K,(R). We shall prove THEOREM 1. Suppose that R is regular and A 0R r is a flat R module. Then the inclusions A A*RF and F A*,R induce a direct sum decomposition

On direct sum decompositions of hestenes algebras

In a *-linear Hestenes algebra, the elements with *-reciprocals are characterized by means of certain direct sum decompositions of the algebra.

Correction to “A characterization of 𝑄𝐹-3 algebras”

J. P. Jans is kind enough to inform me a gap of Necessity proof in my paper appearing in these Proceedings, 13 (1962), 701-703. In this note I shall report Theorem 2 in the paper is however valid by a slight alteration of the proof. In p. 702, the argument between line 9 and line 18 should be replaced by the following: Let ex be a primitive idempotent of A such that 1(N)exXO. Then there exists an element xEL such that 1(N)exx#O for L is faithful. Denote x by axe. + axex, a., ax E A. Since ex( ,,,x age.) C N, I(N)exx = I(N)exaxex and we have 1(N)e,xLex#0. Here, suppose Lex #Aex. Then LexCNex for Nex is the unique maximal left ideal of Aex and it follows 1(N)exLexC1(N)N = 0. But this is a contradiction. Thus we obtain Lex=Aex. Now, let 0 be the epimorphism: L-*Lex(=Aex), defined by 0(x) =xex for all xEL. Since Lex is projective, we have a direct sum decomposition of L:L>DLx', where LxAex. Then as Hom(L, K) is monomorphic to P and Hom(Aex, K) is injective, Hom(Aex, K) is isomorphic to a direct summand of P. Thus if we denote by A the set of all indices X such that 1(N)ex X 0, Hom(:XeA Aex, K) is projective.

On the Homotopy Groups of the Classical Groups

which was conjectured by Serre [2, p. 428]. The theory of [8] showed that the groups in (3) had isomorphic p-primary components for p > 2n, and so we show that this actually holds for p > 2 (it does not hold for p = 2). We prove (1) and (2) by showing that, although the above fibrations do not have a cross-section, there is a map which behaves like a cross-section in homology with coefficients any field of characteristic = 2. If G = su(2n), K = sp(n) or if G = su(2n + 1), K = so(2n + 1) and if a is the automorphism of G leaving K fixed (complex conjugation in the second case, complex conjugation followed by an inner automorphism in the first) then the map in question is the map q: G/K , G given by q(gK) = g(g)-1. Although this map q has been noted previously in the theory of symmetric spaces, its action on cohomology has been determined only recently (in our note 15], which considers general symmetric and other homogeneous spaces). In a later paper we will show that this map is the generalization to symmetric spaces of the characteristic maps for fibre bundles over spheres considered in [9, ?? 23, 24]. We also show that the direct sum decompositions (1) and (2) are simply the decompositions into the +1 and -1 eigenspaces of the map induced

On Symmetric Products and the Steenrod Squares

In his paper Reduced Powers of Cohomology Classes (7) N. E. Steenrod defines a set of invariant cohomology operations. These include as a special case the squares which Steenrod introduced earlier (6). As the definition of the reduced powers depends essentially on a transformation group in the product space K X K X ... X K (p factors) of a complex K it seemed likely that at least in certain cases a close relation between his operations and the Smith-Richardson theory of symmetric products would exist. (See (3), (4) and (5).) In this paper we will discuss the case p = 2, and obtain a definition of the Steenrod squares directly from the exact sequences which are associated with the symmetric product of a space. Our main result can be stated as follows: given a complex X, set K = X X X and let k be the space obtained from K by identifying points which correspond to each other under the involution which exchanges the factors of K. Let 1 be the image in k of the diagonal of K. The Smith theory gives rise to a homomorphism A*:Hf(K; I2) -* H'(k, 1; I2). Further, the cohomology analogue of the Smith-Richardson construction of a basis in Hn(k, 1; I2) leads to the following result: there exists a canonical isomorphism 4' of the direct sum Gn = Z7n' Hi(X; I2) into H2 (k, 1; I2) with A*H2n(K; I2) in its image. Under this isomorphism the element {at}, at = Sqiu; (i = 1, ... , n) u E H'(X), of Gn maps onto A*(u X u). It follows that the Steenrod squares can be defined as the components of the element A*(u X u) in the direct sum decomposition of A*Hf(K) defined by 4,.1 I wish to take this opportunity to thank Professor Steenrod for his helpful remarks and encouragement. 1. Let K be a finite cell complex, t:K -* K an involution (5) of K (i.e. a homeomorphism with period 2 of K onto itself with the following properties: (a) t maps cells onto cells. (b) If a cell is mapped onto itself by t it remains pointwise fixed.) It is a consequence of a, and b, that the set of fixed points under t form a subcomplex L of K. We will denote by k the quotient space K/t, formed by identifying points in K which are t images of each other. p shall denote the projection K -k and we set'l = p(L).

The regular representation of a restricted direct product of finite groups

defined for all f(G)CL2(5). Let W be the smallest self-adjoint algebra of bounded linear operators on L2(5) which is closed in the weak topology for operators (that is, a ring of operators in the sense of [1 ] (1)) and contains the operator R(G) for all GE5O. Let us denote by Z the center of W, that is, the set of those elements of W which commute with every element of W. In accordance with Theorem VII of [5] we can perform the central of L2(65) under W. We obtain a generalized direct sum (=direct integral) decomposition (as defined in [5])