A generalized Frattini subgroup

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

ON THE p-ADIC BINOMIAL SERIES AND A FORMAL ANALOGUE OF HILBERT'S THEOREM 90 – ADDENDUM

The proof of Theorem 3.2 in [1] (P. Tzermias, On the p-adic binomial series and a formal analogue of Hilbert's Theorem 90, Glasgow Math. J.47 (2005), 319–326) contains two opaque claims. The necessary clarifications are provided here.

SUBMODULES OF COMMUTATIVE C*-ALGEBRAS

In this paper we generalise a result of Izuchi and Suárez (K. Izuchi and D. Suárez, Norm-closed invariant subspaces in L∞ and H∞, Glasgow Math. J. 46 (2004), 399–404) on the shift invariant subspaces of $L^\infty(\mathbb{T})$ to the non-commutative setting. Considering these subspaces as $C(\mathbb{T})$-modules contained in $L^\infty(\mathbb{T})$, we show that under some restrictions, a similar description can be given for the ${\mathfrak{B}}$-submodules of ${\mathfrak{A}}$, where ${\mathfrak{A}}$ is a C*-algebra and ${\mathfrak{B}}$ is a commutative C*-subalgebra of ${\mathfrak{A}}$. We use this to give a description of the $\mathbb{M}_n({\mathfrak{B}})$-submodules of $\mathbb{M}_n({\mathfrak{A}})$.

ON WEAK ZIP SKEW POLYNOMIAL RINGS

We introduce the notion of nil(α, δ)-compatible rings which is a generalization of reduced rings and (α, δ)-compatible rings. In [Ore extensions of weak zip rings, Glasgow Math. J.51 (2009) 525–537] Ouyang introduces the notion of right (respectively, left) weak zip rings and proved that, a ring R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip, when R is (α, δ)-compatible and reversible. We extend this result to the more general situation that, when R has (α, δ)-condition and quasi-IFP, then nil (R)[x; α, δ] = nil (R[x; α, δ]); and R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip.

Simplicial cohomology of band semigroup algebras

We establish the simplicial triviality of the convolution algebra l1 (S), where S is a band semigroup. This generalizes some results of Choi (Glasgow Math. J. 48 (2006), 231–245; Houston J. Math. 36 (2010), 237–260). To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on l(S). Crucial to our approach is the use of the structure semilattice of S, and the associated grading of S, together with an inductive normalization procedure in cyclic cohomology. The latter technique appears to be new, and its underlying strategy may be applicable to other convolution algebras of interest.

MONOLITHIC MODULES OVER NOETHERIAN RINGS

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

ON RADICAL FORMULA IN MODULES

We will state some conditions under which if a quotient of a module M satisfies the radical formula of degree k (s.t.r.f of degree k), so does M. Especially, we will introduce some particular modules M′ such that M′ ⊕ M″ s.t.r.f of degree k, when M″ s.t.r.f of degree k. Furthermore, we will show that, under certain conditions, if the completion of a module M s.t.r.f of degree k, then there is a non-negative integer k′ such that M s.t.r.f. of degree k′. Moreover, we state a corrected version of Leung and Man's theorem (K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285–293) on Noetherian rings that satisfies the radical formula.

A CLASS OF EXCHANGE RINGS

It is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

Corrigendum

Proposition 3.1 of the paper Annihilators and the CS-condition, Glasgow Math. J. 40 (1998), 213–222, is incorrect as stated, and consequently the note added in proof is incorrect. Hence the question of Faith and Menal whether every strongly right Johns ring is quasi- Frobenius remains open. The problem is that the assumption that the left socle S1 and the right socle Sr are equal is not established. All we know is that Si⊆Sr = r(J) = l(J) by [8, Lemma 2.2]. We can prove the following result.

Macbeath's Curve and the Modular Group

On p. 244 of Glasgow Math. J.27 (1985) on the right hand side of one of the 6 equations characterizing the 4 fixed points of the involution v a sign error has occurred.The relevant equation should ready0y3y5y6=–1,or the points would not lie on the curve.Correcting the error unfortunately invalidates the model of an elliptic curve given in §6, which therefore has to be re-evaluated. First we find, in the notation of the paper,2 f (x) = ((r + 1)/(R + 2))2.