The theory of unique factorisation in commutative rings has recently been extended to noncommutative Noetherian rings in several ways. Recall that an element x of a ring R is said to be normalif xR = Rx. We will say that an element p of a ring R is (completely) prime if p is a nonzero normal element of R and pR is a (completely) prime ideal. In [2], a Noetherian unique factorisation domain (or Noetherian UFD) is defined to be a Noetherian domain in which every nonzero prime ideal contains a completely prime element: this concept is generalised in [4], where a Noetherian unique factorisation ring(or Noetherian UFR) is defined as a prime Noetherian ring in which every nonzero prime ideal contains a nonzero prime element; note that it follows from the noncommutative version of the Principal Ideal Theorem that in a Noetherian UFR, if pis a prime element then the height of the prime ideal pR must be equal to 1. Surprisingly many classes of noncommutative Noetherian rings are known to be UFDs or UFRs: see [2] and [4] for details. This theory has recently been extended still further, to cover certain classes of non-Noetherian rings: see [3].

In this paper, we present an “order” characterization of completely bounded bimodule maps for bimodules over unital operator algebras. We use this result to prove a bimodule generalization of Wittstock's generalized Hahn-Banach theorem. Our proofs simplify and unify some of Wittstock's arguments.

A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.

A unit regular semigroup [1, 4] is a regular monoid S such that H1 ∩ A(x) ≠ Ø for every xɛS, where H1, is the group of units and A(x) = {y ɛ S; xyx = x} is the set of associates (or pre-inverses) of x. A uniquely unit regular semigroupis a regular monoid 5 such that |H1 ∩ A(x)| = 1. Here we shall consider a more general situation. Specifically, we consider a regular semigroup S and a subsemigroup T with the property that |T ∩ A(x) = 1 for every x ɛ S. We show that T is necessarily a maximal subgroup Hα for some idempotent α. When Sis orthodox, α is necessarily medial (in the sense that x = xαx for every x ɛ 〈E〉) and αSα is uniquely unit orthodox. When S is orthodox and α is a middle unit (in the sense that xαy = xy for all x, y ɛ S), we obtain a structure theorem which generalises the description given in [2] for uniquely unit orthodox semigroups in terms of a semi-direct product of a band with a identity and a group.

In [6] Kulkarni considered the set of values of g for which a given finite group G acts faithfully as a group of orientation-preserving self-homeomorphisms of a compact, connected, orientable surface σg of genus g. Let us denote this set by (G). Then Kulkarni showed that there exists a positive integer Kdepending only on the order d = |G| of G, the exponent e= exp G of G and the structure of a Sylow 2-subgroup G2 of G, satisfying:

Theorem 1. (Kulkarni [6]) (G) consists of all but finitely many non-negative integers g ≡ 1 mod K.

There are two families of group classes that are of particular interest for clearing up the structure of finite soluble groups: Saturated formations and Fitting classes. In both cases there is a unique conjugacy class of subgroups which are maximal as members of the respective class combined with the property of being suitably mapped by homomorphisms (in the case of saturated formations) or intersecting suitably with normal subgroups (when considering Fitting classes). While it does not seem too difficult, however, to determine the smallest saturated formation containing a given group, the same problem regarding Fitting classes does not seem answered for the dihedral group of order 6. The object of this paper is to determine the smallest Fitting class containing one of the groups described explicitly later on; all of them are qp-groups with cyclic commutator quotient group and only one minimal normal subgroup which in addition coincides with the centre. Unlike the results of McCann [7], which give a determination “up to metanilpotent groups”, the description is complete in this case. Another family of Fitting classes generated by a metanilpotent group was considered and described completely by Hawkes (see [5, Theorem 5.5 p. 476]); it was shown later by Brison [1, Proposition 8.7, Corollary 8.8], that these classes are in fact generated by one finite group. The Fitting classes considered here are not contained in the Fitting class of all nilpotent groups but every proper Fitting subclass is. They have the following additional properties: all minimal normal subgroups are contained in the centre (this follows in fact from Gaschiitz [4, Theorem 10, p. 64]) and the nilpotent residual is nilpotent of class two (answering the open question on p. 482 of Hawkes [5]), while the quotient group modulo the Fitting subgroup may be nilpotent of any class. In particular no one of these classes consists of supersoluble groups only.

The braid group on n strings Bn has a presentation as a group with generators σ1, …, σn−1 and relations

In [13] Mazet proved the following result.

If U is an open subset of a locally convex space E then there exists a complete locally convex space (U) and a holomorphic mapping δU: U→(U) such that for any complete locally convex space F and any f ɛ ℋ (U;F), the space of holomorphic mappings from U to F, there exists a unique linear mapping Tf: (U)→F such that the following diagram commutes;

The space (U) is unique up to a linear topological isomorphism. Previously, similar but less general constructions, have been considered by Ryan [16] and Schottenloher [17].

Let X be a compact Hausdorff space, let C(X) denote the algebra of all continuous functions on X, let B be a Banach algebra, and let θ: → C(X) → B be a (possibly discontinuous) homomorphism with dense range. A classical theorem by W. G. Bade and P. C. Curtis ([2, Theorem 4.3]) describes in great detail the structure of θ we shall refer to this result as the Bade–Curtis theorem. Before we give a brief sketch of this theorem, we fix some notation. For Y ⊂ X let I(Y) and J(Y) denote the ideals of all functions in C(X) that vanish on Y and on a neighborhood of Y respectively; if Y = {x} for some x ɛ X, we write mx and Jx for I(Y) and J(Y) respectively. According to the Bade–Curtis theorem there is a finite set {x1,…, xn) ⊂ X, the so-called singularity set of θ, such that θ | ({x1, …, xn}) is continuous. As a consequence, the restriction of θ to the dense subalgebra of C (X) consisting of all those functions which are constant near each Xj (j = 1,…, n) is continuous, and extends to a continuous homomorphism θcont: C(X)→ B. Let θsing: = θ – θcont. Then θsing | I({x1,…, xn}) is a homomorphism onto a dense subalgebra of rad (B). θcont, and θsing are called the continuous and the singular part of θ respectively. Moreover, there are linear maps : C(X)⊒ B such that

(i)

(ii) is a homomorphism, and

Condition (iii) forces the homomorphisms to map into rad(B); such homomorphisms are called radical homomorphisms.

Let V be a regular semigroup and an ideal extension of a semigroup S by a semigroup Q Congruences on V can be represented by triples of the form (σ, P, τ), here called admissible, where a is a congruence on S, P is an ideal of Q and τ is a O-restricted congruence on Q/P satisfying certain conditions. We characterize the trace relation T on V in terms of admissible triples. When the extension V of S is strict, for a congruence v on V given in terms of an admissible triple, we characterize vK, vK, vT and vT again in terms of admissible triples.

A group is called metacyclic in case both its commutator subgroup and commutator quotient group are cyclic. Thus a metacyclic group is a cyclic extension of a cyclic group, and metacyclic groups are among the best understood of the nonabelian groups. Many interesting groups are metacyclic. For instance, the dihedral groups and the “odd” dicyclic groups are metacyclic; see [4, pp. 9–11] for more examples. Here we shall consider the actions of these groups on bordered Klein surfaces.

Throughout the paper we consider only finite groups.

J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.

One of the most important results of operator theory is the spectral theorem for normal operators. This states that a normal operator (that is, a Hilbert space operator T such that T*T= TT*), can be represented as an integral with respect to a countably additive spectral measure,

Here E is a measure that associates an orthogonal projection with each Borel subset of ℂ. The countable additivity of this measure means that if x Eℋ can be written as a sum of eigenvectors then this sum must converge unconditionally.

Let V and W be two vector spaces over the field of real numbers R. Then we have the notion of the tensor product V ⊗ W. If V and W are inner product spaces with their inner products given respectively by «,»v and «,» w, then V ⊗ W is also an inner product space with inner product denned by

Let Em denote the m-dimensional Euclidean space with the canonical Euclidean inner product. Then, with respect to the inner product defined above, Em ⊗Em is isometric to Em. By applying this algebraic notion, we have the notion of tensor product mapf ⊗h: M→ E: M ⊗= Em; associated with any two maps f: M→Em and h:M→E of a given Riemannian manifold (M, g) defined as follows:

Denote by R(M) the set of all transversal immersions from an n-dimensional Riemannian manifold (M, g) into Euclidean spaces; i. e., immersions f:M→Em with f(p) ∉T*(TPM) for p ∈ M. Then ⊗ is a binary operation on R(M). Hence, if f: Mm and h: M→Em are immersions belonging to R(M), then their tensor product map f ⊗ h: M→ Em ⊗ Em ≡ Emm is an immersion in R(M), called the tensor product immersionof f and h.