Let K be a field of characteristic different from 2, and a, b quadratically independent elements of K. Put J= K(√a, √b). In [4], Jensen and Yui discuss the question of quaternionic (Q8) extensions of J, and give a survey of known results. In [8], Ware discusses (among other things) some general conditions for, and relations between, the existence of Q8 and D4 (dihedral) extensions of K. A general theorem of Witt [9] says that J will have a quaternionic extension J(√u) if and only if there exists a 3 × 3 matrix P over K such that PPt = diag(a, b, 1/ab), and an appropriate value for u is given in terms of the entries of P. The problem of actually finding P in a particular case is not trivial.

Let G be a group. The norm, or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G. In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω(G). The Wielandt series of subgroups ω1(G) is defined by: ω1(G) = ω(G) and for i ≥ 1, ωi+1(G)/ ω(G) = ω(G/ωi, (G)). The subgroups of the upper central series we denote by ζi(G).

A locally convex space (E, ) has the Mazur Property if and only if every linear -sequential continuous functional is -continuous (see [11]).

In the Banach space setting, a Banach space X is a Mazur space if and only if the dual space X* endowed with the w*-topology has the Mazur property. The Mazur property was introduced by S. Mazur, and, for Banach spaces, it is investigated in detail in [4], where relations with other properties and applications to measure theory are listed. T. Kappeler obtained (see [8]) certain results for the injective tensor product and showed that L1(μ, X), the space of Bochner-integrable functions over a finite and positive measure space (S, σ, μ), is a Mazur space provided X is also, and ℓ1 does not embed in X.

Many of the classical inequalities of analysis can be written in the form P(x) ≥ 0 for x ∈ I or P(x) > 0 for x ∈ I′, where P(x) is a polynomial and I′ ⊂ I are certain intervals on the real line. This gives rise to the question of where the zeros of P(x) are located. For example, if f is a polynomial with real zeros, then an inequality of Laguerre [8, p. 171 f.] asserts that

for all x. A detailed study of the zeros of this particular P(x) has been made [5].

The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right R-module if and only if R is right artinian and every indecomposable projective right R-module is uniform and weakly R-injective. We show that in the above characterization the requirement that indecomposable projective right R-modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right QI-rings. A ring R is said to be a right QI-ring if every quasi-injective right R-module is injective. QI-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others.

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].

The present note deals with bounded endomorphisms of free p-algebras (pseudocomplemented lattices). The idea of bounded homomorphisms was introduced by R. McKenzie in [8]. T. Katriňák [5] subsequently studied the properties of bounded homomorphisms for the varieties of p-algebras. This concept is also an efficient tool for the characterization of, so-called, splitting as well as projective algebras in the varieties of all lattices or p-algebras. For details the reader is referred to [2], [5], [6], [7] and other references therein. Let us emphasize that the main results that are contained in the above mentioned references strongly depend on the boundedness of each endomorphism of any finitely generated free algebra in a given variety.

The existence of r-regular graphs such that each edge lies in exactly t triangles, for given integers t < r, is studied. If t is sufficiently close to r then each such connected graph has to be the complete multipartite graph. Relations to graphs with isomorphic neighborhoods are also considered.

An associative ring R is called a left SI-ring if every singular left R-module is injective. In Goodearl [4] it is shown that these rings have a finite ring decomposition into a ring K with K/Soc K left semisimple, and simple rings which are Morita equivalent to left SI-domains.

Ramanujan's short third notebook contains 33 pages of unorganized material. On page 386 of the third notebook, in the pagination of the Tata Institute's publication [3], Ramanujan offers the following remarkable identity.

We have studied relationships between almost relative projectivity and Nakayamaings [8]. In this paper we shall further investigate certain characterizations of right Nakayama rings in terms of almost relative projectives (or injectives). We shall consider three conditions (A), (B) and (C) (see Section 1), which are always satisfied for the relative projective modules, but not for almost relative projectives in general. As an application of [9, Theorem] and [10, Theorem 2], we shall show that a right artinian ring is right Nakayama if and only if one of the above three conditions holds true for almost relative projectives (Corollary to Theorem 1). Moreover we shall give a characterization of two-sided Nakayama rings related to (C) and the dual (C#) (Theorem 2). Finally we shall investigate the transitivity of almost relative projectives, which is the converse of (B), and give some characterizations of right Nakayama rings related to the transitivity.

The purpose of this short note is to give a new and shorter proof of the following theorem of Johnson [1], and to extend it somewhat.

Theorem 1. Let G be a finite non-cyclic p-group possessing a non-empty subset X such that, for each x in X, <X/{x}>G′ is a complement for <x> in G. Then the Schur multiplier of G is non-trivial.

Let X be a complex Banach space and H be a hermitian operator on X. Then in [7] Sinclair proved that r(H) = ¶H¶, where r(H) and ¶H¶ are the spectral radius and the operator norm of H, respectively.

For a commuting n-tuple T = (T1,…, Tn) of operators on X, we denote the (Taylor) joint spectrum of T by σ(T) (see [9]) and define the joint operator norm ¶T¶ and the joint spectral radius r(T) by

and

respectively.

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

The notation in this paper will be standard and it may be found in [2] or [8]. Throughout the paper, the notation A ⊂' B will mean that A is an essential submodule of the module B. Given an arbitrary ring R and R-modules M and N, we say that M is weakly N-injective if and only if every map φ:N → E(M) from N into the injective hull E(M) of M may be written as a composition σ〫 , where :N→M and σ:M→E(M) is a monomorphism. This is equivalent to saying that for every map φ:N→E(M), there exists a submodule X of E(M), isomorphic to M, such that φ(N) is contained in X. In particular, M is weakly R-injective if and only if, for every x ∈ E(M), there exists X ⊂ E(M) such that x ∈ X ≌ M. We say that M is weakly-injective if and only if it is weakly N-innjective for every finitely generated module N. Clearly, M is weakly-injective if and only if, for every finitely generated submodule N of E(M), there exists X ⊂ E(M) such that N ⊂ X ≌ M.

In [3], Sharp and Taherizadeh introduced concepts of reduction and integral closure of an ideal I of a commutative ring R (with identity) relative to an Artinian R-module A, and they showed that these concepts have properties which reflect some of those of the classical concepts of reduction and integral closure introduced by Northcott and Rees in [2].

Let M2 be a (connected) surface in Euclidean 3-space E3, and let G:M2→S2(1) ⊂ E3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M2 is a surface of constant mean curvature if and only if, as a map from M2 to S2(1), G is harmonic, or equivalently, if and only if

where δ is the Laplace operator on M2 corresponding to the induced metric on M2 from E3 and where G is seen as a map from M2to E3. A special case of (1.1) is given by

i.e., the case where the Gauss map G:M2→E3 is an eigenfunction of the Laplacian δ on M2.

In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X. If F is a closed set in the plane C, let X(T, F) consist of all y ∈ X satisfying thes identity

where f:C\F → X is analytic. It is then easy to see that X(T, F) is a T-invariant linear manifold in X. Moreover, if y ∈ X then

is a compact subset of the spectrum σ(T). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X(T, F) and γ(y, T). Recall that T is decomposable if whenever G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy is called the local spectrum σ(y, T) and coincides with γ(y, T). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f:V → X satisfying (z − T)f(z) = 0 on V. If T is decomposable and the restriction T | X(T, F) is also decomposable for each closed F, then T is called strongly decomposable. We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space.

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:

(i) biprojectivity of N(E);

(ii) biflatness of N(E);

(iii) homological finite-dimensionality of N(E);

(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

The previous results on isometrically characterizing ln∞ in terms of operator ideal norms can be summarized as follows.

Let E be an n-dimensional Banach space.

(1) If λ(E) = 1, then (see [3], [5], [6]), where λ(E) is the projection constant of E (see [4]).

(2) If π(E) = n, then (see [1],[2]).

(3) If Δ2(E) = √n, then (see [8]), where Δ2(E) is the 2-dominated constant of E(see [4]).

(4) If for any linear operator T: → E, v1(T) = (T), then (see [7]), where v1 is the 1-nuclear norm (see [4]).