Abstract: Let Σ be a commutative associative ring with unit, let R={a α } be some set of symbols, and let \( \mathfrak{A}_{\Sigma R} \) be the free associative algebra over Σ with free generating set R. In the ring \( \mathfrak{A}_{\Sigma R} \), the set R generates a Lie subring \( \mathfrak{A}_{\Sigma R}^{\left( - \right)} \) with respect to the operations x o y=xy−yx, addition, and scalar multiplication by elements of Σ.

Abstract: Provided is a polymer compound containing a constituent unit having a group represented by formula (1). [In the formula, the ring R 1A and the ring R 2A are each an aromatic hydrocarbon ring or a heterocyclic ring, and these rings may have substituent groups. nA is an integer between 0 and 5, and nB is an integer between 1 and 5. L A and L B are each an alkylene group, a cycloalkylene group, an arylene group, a divalent heterocyclic group, a group represented by -NR'-, an oxygen atom or a sulfur atom. R' is a hydrogen atom, an alkyl group, or the like. Q 1 is a crosslinking group.]

Abstract: Let C be a commutative ring with 1. Let A be a coalgebra over C with diagonal map d : A ? A ? C A (it is assumed A has a counit ? : A ? C) and let R be an algebra over C with multiplication m: R ? C R ? R (it is assumed R has a unit p : C ? R).

Abstract: Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative categories and study the relations between the obtained algebras. The associative case is richer since it admits the notion of unit element. We use this fact to find sufficient conditions for hom-associative algebras to be associative and classify the implications between the hom-associative types of unital algebras.

Abstract: A complete list of homogeneous operators in the Cowen-Douglas class $$B_n({\mathbb{D}})$$ is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.

Abstract: Using the Luthar-Passi method, we investigate the Zassenhaus and Kim- merle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held sim- ple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture. Let U(ZG) be the unit group of the integral group ring ZG of a finite group G. It is well known that U(ZG )= U(Z) × V (ZG), where V (ZG )= g∈G αgg ∈ U(ZG) | g∈G αg =1 ,α g ∈ Z.

Abstract: Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup $S$ that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.

Abstract: Using the Luthar--Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture.

Abstract: The invention provides a method and a device for outputting rings of a mobile terminal, wherein the method comprises the following steps of: 1, in response to a storage instruction input by a user, pre-storing a plurality of rings that can be selected; 2, selecting at random a first ring from the plurality of pre-stored rings to output when a ring triggering event occurs, and the selected ring is differed from the ring that is selected and output in response to the last ring triggering event; wherein the inventive device comprises: a ring storage unit for storing the plurality of rings that can be selected and used in response to the storage instruction input by the user; a random ring-outputting unit for selecting at random a first ring from the plurality of rings pre-stored in the ring storage unit to output when the ring triggering event occurs, and the selected ring is differed from the ring that is selected when the last ring triggering event occurs. By using the invention, it is possible to randomly select the rings to output in case of occurrence of the ring triggering events, such as incoming calls, incoming short messages, alarm clock or arrival of schedule and so on, which avoids the tedious output rings.

Abstract: Let $KG$ denote the group algebra of the group $G$ over the field $K$ and let $U(KG)$ denote its group of units. Here without the use of a computer we give presentations for the unit groups of all group algebras $KG$, where the size of $KG$ is less than 1024. As a consequence we find the minimum counterexample to the Isomorphism Problem for group algebras.

Abstract: Let R be an associative ring with identity. An element a∈R is called clean if a=e+u with e an idempotent and u a unit of R and a is called strongly clean if, in addition, eu=ue. A ring R is called clean if every element of R is clean and R is strongly clean if every element of R is strongly clean. In the paper [Nicholson and Zhou, Clean rings: a survey, Advances in Ring Theory, 181–198, World Sci. Pub., Hackensack, NJ, 2005], the authors brought out an up to date account of the results in the study of clean rings. Here, we give an account of the results on strongly clean rings.

Abstract: An apparatus for 3D mesh compression based on quantization, includes a data analyzing unit (510) for decomposing data of an input 3D mesh model into vertices information (511 ) property information (512) representing property of the 3D mesh model, and connectivity information (515) between vertices constituting the 3D mesh model: and a mesh model quanitzing unit (520) for producing quantized vertices and property information of the 3D mesh model by using the vertices, property and connectivity information (511, 512, 513). Further, the apparatus for 3D mesh compression based on quantization includes a decision bit encoding unit (535) for calculating a decision bit by using the quantized connectivity information and then encoding the quantized vertex information, property information and connectivity information (511, 512, 513) by using the decision bit.

Abstract: Let R be a ring with identity. An element in R is said to be clean if it is the sum of a unit and an idempotent. R is said to be clean if all of its elements are clean. If every idempotent in R is central, then R is said to be abelian. In this paper we obtain some conditions equivalent to being clean in an abelian ring.

Abstract: Let R be a ring and M(R) the set consisting of zero and primitive idempotents of R. We study the rings R for which M(R) is multiplicative. It is proved that if R has a complete finite set of primitive orthogonal idempotents, then R is a finite direct product of connected rings precisely when M(R) is multiplicative. We prove that if R is a (von Neumann) regular ring with M(R) multiplicative, then every primitive idempotent in R is central. It is also shown that this does not happen even in semihereditary and semiregular rings. Let R be an arbitrary ring with M(R) multiplicative and e ∈ R be a primitive idempotent, then for every unit u ∈ R, it is proved that eue is a unit in eRe. We also prove that if M(R) is multiplicative, then two primitive idempotents e and f in R are conjugates, i.e., f = ueu −1 for some u ∈ U(R), if and only if ef ≠ 0.

Abstract: We show that the free product of any collection of non-trivial unital l-groups with fixed strong unit exists. Equivalently, the free product of any collection of non-trivial generalized MV-algebras exists. We then investigate free products in some of the varieties of these algebras.

Abstract: There are two molecules in the asymmetric unit of, C10H8N2O, with dihedral angles between the aromatic ring planes of 75.9 (1) and 79.3 (1)°.

Abstract: Caradus [2] has defined an operators Τ G B(X) on a Banach space to be "generalised Fredholm" if Τ = Τ ST has a generalised inverse S G B (X) for which I ST Τ S is Fredholm, and in a sequence of papers Schmoeger has extended the idea to "generalised invertible" Banach algebra elements ([7; 8]) and elements which are generalised Fredholm relative to the socle ([6]). In this note we look at elements which are "generalised T-Fredholm" , where Τ : A » Β is a unital homomorphism of Banach algebras. Let A and Β be complex algebras with unit e, and suppose that Τ : A > Β is an algebra homomorphism in the sense that T(ab) = T(a)T(b) and T(e) = e. Recall that an element a G A is invertible provided there is an element b such that ab = ba = e. We will denote by A~l the set of all invertible elements of A, and by C the complex plane. If a G -A, the spectrum of a is the set σ^(α) = {λ G C : 'e-a<£A-1}. An element a G A is relatively regular if there is b G A such that a = aba. We call such b a generalised inverse of a. We will denote the set of all relatively regular elements of A by A. A subspace J of A is a left (right) ideal if ab G J (ba G J) whenever a G A and b G J. An ideal in A is a subspace that is simultaneously a left and a right ideal in

Abstract: Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R, and a is called strongly clean if, in addition, eu = ue. A ring R is clean if every element of R is clean, and R is strongly clean if every element of R is strongly clean. When is a matrix ring over a strongly clean ring strongly clean? Does a strongly clean ring have stable range one? For these open questions, we prove that 𝕄 n (C(X)) is strongly π-regular (hence, strongly clean) where C(X) is the ring of all real valued continuous functions on X with X a P-space; C(X) is clean iff it has stable range one; and a unital C*-algebra in which every unit element is self-adjoint is clean iff it has stable range one. The criteria for the ring of complex valued continuous functions C(X,ℂ) to be strongly clean is given.

Abstract: In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these two properties to lattice-ordered groups. An example is constructed to illustrate that the two properties are distinct, which solves a question from [17]. We also investigate these properties with respect to archimedean l-groups with weak order unit, as well as commutative semiprime f-rings.

Abstract: The asymmetric unit of the title compound, C16H12O3, contains two crystallographically independent mol­ecules. The isochromene ring system is planar (maximum deviation 0.024 A) and is oriented at dihedral angles of 2.63 (3) and 0.79 (3)° with respect to the methoxy­benzene rings in the two independent mol­ecules.

Abstract: Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with n ≥ 3. We denote by M n (R) the ring of all n × n matrices over R. Let 〈 $$ \Im _n $$ (R)〉 be the additive subgroup of M n (R) generated additively by all idempotent matrices. Let $$ \mathfrak{V} $$ = 〈 $$ \Im _n $$ (R)〉 or M n (R). We describe the additive preservers of idempotence from $$ \mathfrak{V} $$ to M m (R) when 2 is a unit of R. Thereby, we also characterize the Jordan (respectively, ring and ring anti-) homomorphisms from M n (R) to M m (R) when 2 is a unit of R.

Abstract: We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.

Abstract: The present invention relates to a rotating ring apparatus comprising: -a stationary ring (1), -a rotating ring (2) for rotation around a central region (3), -a rotation unit (4, 5) for radially bearing and rotating said rotating ring (2) with respect to the stationery ring (1), said rotation unit including a stationary induction element (4) mounted on said stationery ring (1) and a rotating induction element (5) mounted on said rotating ring (2), and -a controller (8) for controlling said rotation unit (4, 5) to rotate and levitate the rotating induction element (5).

Abstract: The study of the projective coordinate ring of the (geometric invariant theory) moduli space of n ordered points on P^1 up to automorphisms began with Kempe in 1894, who proved that the ring is generated in degree one in the main (n even, unit weight) case. We describe the relations among the invariants for all possible weights. In the main case, we show that up to the symmetric group symmetry, there is a single equation. For n not 6, it is a simple quadratic binomial relation. (For n=6, it is the classical Segre cubic relation.) For general weights, the ideal of relations is generated by quadratics inherited from the case of 8 points. This paper completes the program set out in [HMSV1].