Narrow Results

A right R-module M_R over any ring R with 1 is called torsion-free if it satisfies the equality for every r∈R. An equivalent definition was used by Hattori [11]. We establish various properties of this concept, and investigate rings (called torsion-free rings) all of whose right ideals are torsion-free. In a torsion-free ring, the right annihilators of elements are always idempotent flat right ideals. The right p.p. rings are characterized as torsion-free rings in which the right annihilators of elements are finitely generated. An example shows that torsion-freeness ness is not a Morita invariant.

Several ring and module properties are proved, showing that, in several respects, torsion-freeness ness behaves like flatness. We exhibit examples to point out that the concept of torsion-freeness ness discussed here is different from other notions.

In this paper, by taking the class of all $C3$ (or $D3$) right $R$-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via $D3$-covers (or $D3$-envelopes) and a right $V$-ring (or a right noetherian $V$-ring) via $C3$-covers (or $C3$-envelopes). By using isosimple-projective preenvelope, we obtained that if $R$ is a semiperfect right FGF ring (or left coherent ring), then every isosimple right $R$-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.

One of the most frequent sitting styles of Asians in everyday life is a cross-legged sitting. The cross-legged sitting results in higher compression load in spine than sitting on a chair, so a proper sitting posture is more needed. The purpose of this study was to classify the spinal posture during cross-legged sitting from the seat pressure pattern for future usage in the posture monitoring system. Twenty young men participated in this study. The seat pressure was measured for three spinal postures of flat, slump, and lordosis when subjects were instructed to pose a certain posture while seated on the floor with legs crossed. The contact area was divided into feet and buttocks by using a filter with a pressure threshold (${\text{th}}_f$). A decision tree was developed for the classification of three postures, with a decision variable of feet to buttocks pressure ratio. The three spinal postures were classified by comparison of feet-buttocks ratio ($R_{fb})$ and thresholds (${\text{th}}_{R1}$, ${\text{th}}_{R2}$): a slump posture with a greater $R_{fb}$ than ${\text{th}}_{R1}$, a lordosis posture with a smaller $R_{fb}$ than ${\text{th}}_{R2}$. Each threshold was calculated by adding or subtracting a certain percentage ($\alpha ,\beta$) to or from the $R_{fb}$ of flat posture and the classification accuracy was investigated with a range of thresholds. The accuracy of classification achieved 99.38% for certain ranges of thresholds. The developed algorithm showed the best performance when $\alpha$ and $\beta$ were in the range of 2.85–5.67% and 1.58–2.20%, respectively. The feet-buttocks pressure ratio showed significant correlation with lumbar angle ($r=0.67$, $p<0.001$). Anterior and posterior tilts of upper body in the slump and lordosis postures would result in more pressure concentration in the feet and buttocks, respectively, which was incorporated in the classification algorithm of this study. The result of this study could be extended to the real-time or offline monitoring of the sitting posture.

An act A_S is called torsion free if for any a, b ∈ A_S and for any right cancellable element c ∈ S the equality ac = bc implies a = b. In [M. Satyanarayana, Quasi- and weakly-injective S-system, Math. Nachr.71 (1976) 183–190], torsion freeness is considered in a much stronger sense which we call in this paper strong torsion freeness and will characterize monoids by this property of their (cyclic, monocyclic, Rees factor) acts.

In this paper, we investigate the notion of dc-po-flat $S$-posets as the ones for which the associated tensor functors transfer merely down-closed embeddings (embeddings with down-closed images in codomains) to embeddings. We investigate derived flatness notions in regard to dc-po-flatness in parallel with po-flatness notions and give examples to clarify new notions and their implications. As the characterization of flat acts by means of embeddings into cyclic acts, stated by Fleischer, is not valid for $S$-posets, it eventuates in introducing the new notion of cyclical po-flatness, situated strictly between weak po-flatness and po-flatness, though, we express a counterpart characterization for dc-po-flatness. At the end, we expose relationships between some po-flatness properties and regular injectivity.

Let ${𝒜}$ and $𝔄$ be two Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible actions of $𝔄$ on ${𝒜}$. Let ${𝒜}⋊𝔄$ be a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. In this paper, we investigate some homological aspects such as injectivity, projectivity and flatness of ${𝒜}⋊𝔄$ and give some necessary and sufficient conditions for injectivity, projectivity and flatness of ${𝒜}⋊𝔄$.

Let G = ℤ/2ℤ ≀ ℤ be the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1 – x is invertible, with x a generator of ℤ ⊂ G. Then R is not flat over kG. If k = ℂ, this applies in particular to the algebra of unbounded operators affiliated to the group von Neumann algebra of G.

We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.