Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

J. Wichman has asked about semisimple radical classes of involution algebras. In the present paper we describe the semisimple radical classes of involution algebras over a field K* with involution *. If K* is infinite, then there are only trivial semisimple radical classes. If K* is finite then these classes are subdirect closures of strongly hereditary finite sets of finite idempotent algebras. In proving this result we determine the structure of certain simple involution algebras. We prove that the variety of symmetric involution algebras over Z(2) does not have attainable identities, answering a problem posed by Gardner [2]. Most of the results are valid also for involution rings (over the integers).

In [1] the first author considered the following Engel-like condition for a pair of elements x, y of a group G.

There exist r = r(x, y)≧0 and d = d(x, y)≧1 such that[x,ry] = [x,r+dy]. (*)

In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an “octad generator”; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.

All our rings will be commutative with identity not equal to zero. Also R will always denote a ring. is a filter of ideals of R if is a nonempty set of ideals of R satisfying: I∈ and J is an ideal of R with I⊂J, then J∈, and if I, J∈ then I∩J∈. A Gabriel topology of R is a filter of ideals of R satisfying: if J∈ and I is an ideal of R with (I:x)∈ for all x∈J, then I∈. See the B. Stenström text [6]. We say that a ring R is an FGC ring if every finitely generated R-module is a direct sum of cyclic R-modules. Use mspec R for the set of all maximal ideals of R.

Throughout this paper all spaces are T1 and N will denote the set of all positive integer numbers.

A quasi-metric on a set X is a non-negative real-valued function d on X × X such that, for all x, y, z ∈ X, (i) d(x, y) = 0 if, and only if, x = ysemicolon (ii) d(x, y)≦d(x, z) + d(z, y).

Let T2 = {(eix1, eix2):0 ≦ xj<2π, j=1,2} be a two dimensional torus and r, s, t and k be positive integers with k>r+s+t–2. Our main object is to study the approximation and interpolation properties of a class of smooth functions whose restrictions to each triangle of a three direction mesh lie in the linear span of or 0≦μ≦r–1, r+s–l≦μ+ν≦r+s+t–2, or 0≦ν≦s–1, r+s–1≦μ+ν≦r+s+t–2} Where (z1, z2) ∈ T2.

The concept of “root vectors” is investigated for a class of multiparameter eigenvalue problems

where operate in Hilbert spaces Hm and . Previous work on this “uniformly elliptic” class has demonstrated completeness of the decomposable tensors x1 ⊗…⊗ xk in a subspace G of finite codimension in H=H1 ⊗…⊗ Hk, but questions remain about extending this to a basis of H. In this work, bases of elements ym, in general nondecomposable but satisfying recursive equations of the type are constructed for the “root subspaces” corresponding to λ∈ℝk.

It is well known that a ring R is semiprime Artinian if and only if every right ideal is an injective right R-module. In this paper we shall be concerned with the following general question: given a ring R all of whose right ideals have a certain property, what implications does this have for the ring R itself? In practice, it is not necessary to insist that all right ideals have the property, usually the maximal or essential right ideals will suffice. On the other hand, Osofsky proved that a ring R is semiprime Artinian if and only if every cyclic right R-module is injective. This leads to the second general question: given a ring R all of whose cyclic right R-modules have a certain property, what can one say about R itself?

Throughout the paper, the symbols G1 and G2 will denote two locally compact abelian groups with character groups X1 and X2, respectively. Haar measures on Gj are denoted by μj; the ones on Xj are denoted by θj (j=1,2). The measures μj and θj are normalized so that the Plancherel Theorem holds (see [7, p. 226, Theorem 31.18]).

Let (X, Σ,μ) be a measure space (where μ is a positive σ-additive measure) and let Lp(X,Σ,μ), 1≦p≦ + ∞ be the usual real Banach lattices.

In his book [1] Divinsky refers to eight radicals as classical. In [6] radicals were considered such that the radical of each one-sided ideal of a ring may be expressed as the intersection of a left ideal and a right ideal of the ring. From results obtained there it was deduced that seven of these eight radicals have this property. The purpose of this note is to give a proof that this property also holds for the remaining one of these classical radicals.

In [1] J. Ax studied a class of fields with similar properties as finite fields called pseudo-finite fields. One can prove that pseudo-finite fields are precisely the infinite models of the first-order theory of finite fields. Similarly a near-field F is called pseudo-finite if F is an infinite model of the first-order theory of finite near-fields. The structure theory of these near-fields has been initiated by U. Feigner in [5].

If G is an additive (but not necessarily abelian) group and S is a semigroup of endomorphisms of G, the endomorphism near-ring R of G generated by S consists of all the expressions of the form ɛ1s1+…+ɛnsnwhere ɛi=±1 and si∈S for each i. When functions are written on the right, R forms a distributively generated left near-ring under pointwise addition and composition of functions. A basic reference on near-rings which has a substantial treatment of endomorphism near-rings is [6].

Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal if S° contains a unique inverse of each element of S. An inverse transversal S° of S is called multiplicative if x°xyy° is an idempotent of S° for every x, y∈S, where x° denotes the unique inverse of x∈S in S°. In Section 1, we obtain a necessary and sufficient condition in order for inverse transversals to be multiplicative.

Kurosh-Amitsur radical theories have been developed for various algebraic structures. Whenever the notion of a normal substructure is not transitive, this causes quite some problems in obtaining satisfactory general results. Some of the more important questions concerning the general theory of radicals are whether semisimple classes are hereditary, do radical classes satisfy the ADS-property, can semisimple classes be characterized by closure conditions (e.g., is semisimple=coradical), is Sands' Theorem valid and lastly, does the lower radical construction terminate. For associative and alternative rings, all these questions have positive answers. The method of proof is the same in both cases. In [15], Puczylowski used the results of Terlikowska-Oslowska [18, 19] and hinted at a condition which is crucial in obtaining the positive answers to the above questions.

Zemanian [17] obtained abelian theorems for the Hankel and K-transforms of functions and then extended his results to the corresponding transforms of distributions in the sense of Schwartz [11]. Jones [6] has discussed at length asymptotic behaviours of transforms generalized in his sense. Following the technique of Zemanian many authors have obtained abelian theorems for more general transforms of functions and distributions in the sense of Schwartz. Mention may be made of the works of Joshi and Saxena [7], Lavoine and Misra [8] and Pathak [10]. However, these authors were confined to the transforms of real variables only.

Let S be a regular semigroup. An inverse subsemigroup S° of S is an inverse transversal if |V(x)∩S°| = 1 for each x∈S, where V(x) denotes the set of inverses of x. In this case, the unique element of V(x)∩S° is denoted by x°, and x°° denotes (x°)–1. Throughout this paper S denotes a regular semigroup with an inverse transversal S°, and E(S°) = E° denotes the semilattice of idempotents of S°. The sets {e∈S:ee° = e} and {f∈S:f°f=f} are denoted by Is and Λs, respectively, or simply I and Λ. Though each element of these sets is idempotent, they are not necessarily sub-bands of S. When both I and Λ are sub-bands of S, S° is called an S-inverse transversal. An inverse transversal S° is multiplicative if x°xyy°∈E°, and S° is weakly multiplicative if (x°xyy°)°∈E° for every x, y∈S. A band B is left [resp. right] regular if e f e = e f [resp. e f e = f e], and B is left [resp. right] normal if e f g = e g f [resp. e f g = f e g] for every e, f, g∈B. A subset Q of S is a quasi-ideal of S if QSQ ⊆ S.

In each metric space (X, d) there is defined the space Lip X of complex-valued, bounded, and uniformly Lipschitzian functions. In the algebra Lip X, it is natural to ask for ideals closed in various notions of convergence, and also to identify the invertible elements. In particular, are the invertible elements exactly those with no zero in X? Wiener's Tauberian Theorem in Fourier analysis is the first and most remarkable example of this harmonious state of affairs. A moment's reflection confirms that, for the algebra Lip X, this is true only for compact metric spaces X, the trivial examples in our investigation. We therefore introduce a type of convergence weaker than convergence in norm; it has already proved useful in some problems in descriptive set theory and reflects in a subtle way the metric properties of X. A sequence (fn) in Lip X converges strongly to g, written s – limfn=g, if ∥fn∥≦C in the Banach space Lip X and lim fn(x)=g(x) for each element x of X. In Section 3 we explain how this is really a type of convergence in the dual space of a certain Banach space . This brings us to the edge of some recondite questions about iterated (or even transfinite) limits, and we have adhered to the notion of strong limits to avoid these questions. To illustrate the differences between these two approaches, we mention this problem: which maximal ideals of Lip X are closed with respect to strong convergence of sequences? This is not the problem studied in Section 1.

Given any subspace N of a Banach space X, there is a subspace M containing N and of the same density character as N, for which there exists a linear Hahn–Banach extension operator from M* to X*. This result was first proved by Heinrich and Mankiewicz [4, Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Löwenheim–Skolem theorem due to Stern [11], which in turn relies on the Löwenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [7], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model Theory.