Hard to summarize concisely; here are the high points. The first two statements below are ring-theoretic; in these R is a nontrivial ring, R^\omega, and \bigoplus_\omega R are the direct product, respectively direct sum, of countably many copies of R; the remaining two statements are in the context of general algebra (a.k.a. universal algebra): (i) There exist nontrivial rings R for which one has surjective homomorphisms \bigoplus_\omega R -> R^\omega -- but in such cases, R^\omega is in fact finitely generated as a left R-module. (ii) There exist nontrivial rings R for which one has surjective homomorphisms R^\omega -> \bigoplus_\omega R -- but in such cases, R must have DCC on finitely generated right ideals. (iii) The full permutation group S on an infinite set \Omega has the property that the |\Omega|-fold direct product of copies of S is generated over its diagonal subgroup by a single element. (iv) Whenever an algebra S in the sense of universal algebra has the property that the countable direct product S^\omega is finitely generated over its diagonal subalgebra (or even when the corresponding property holds with an ultrapower in place of this direct product), S has some of the other strange properties known to hold for infinite symmetric groups (cf. math.GR/0401304).
This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation \[\Delta = .\] I show that it acts transitively on a 3-regular tree, and that $\Gamma=< b,b^{t^{-1}}$ stabilizes a vertex and acts by restriction on a binary rooted tree. $\Gamma$ is a "fractal group", generated by a 3-state automaton, and is a discrete analogue of the monodromy action of iterates of f(z)=z^2-1 on associated coverings of the Riemann sphere. $\Delta$ shares many properties with the Thompson group $F$. The proof of the main result (amenability of $\Delta$) is incomplete in the present form; please refer to the paper arxiv.org/math.GR/0305262, joint with Balint Virag, for a complete proof.
Abstract Let g be a finite-dimensional real Lie algebra. Let ρ : g → End (V) be a representation of g in a finite-dimensional real vector space. Let C V =( End (V)⊗S( g )) g be the algebra of End(V)-valued invariant differential operators with constant coefficients on g . Let U be an open subset of g . We consider the problem of determining the space of generalized functions φ on U with values in V which are locally invariant and such that C V φ is finite dimensional. In this article we consider the case g = sl (2, R ) . Let N be the nilpotent cone of sl (2, R ) . We prove that when U is SL(2, R ) -invariant, then φ is determined by its restriction to U ⧹ N where φ is analytic (cf. Theorem 6.1). In general this is false when U is not SL(2, R ) -invariant and V is not trivial. Moreover, when V is not trivial, φ is not always locally L1. Thus, this case is different and more complicated than the situation considered by Harish-Chandra (Amer. J. Math 86 (1964) 534; Publ. Math. 27 (1965) 5) where g is reductive and V is trivial. To solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone N . We do this locally in a neighborhood of a nilpotent element Z of g (cf. Theorem 4.1) and on an SL(2, R ) -invariant open subset U ⊂ sl (2, R ) (cf. Theorem 4.2). Finally, we also give an application of our main theorem to the Superpfaffian (Superpfaffian, prepublication, e-print math.GR/0402067, 2004).
Malak Alnimer, Khaldoun Al-Zoubi, Mohammed Al-Dolat
<abstract><p>Let $ \Gamma $ be a group, $ \mathcal{A} $ be a $ \Gamma $-graded commutative ring with unity $ 1, $ and $ \mathcal{D} $ a graded $ \mathcal{A} $-module. In this paper, we introduce the concept of graded weakly $ J_{gr} $-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly $ J_{gr} $ -semiprime submodules. For example, we give a characterization of graded weakly $ J_{gr} $-semiprime submodules. Also, we find some relations between graded weakly $ J_{gr} $-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly $ J_{gr} $-semiprime submodules are investigated. A proper graded submodule $ U $ of $ \mathcal{D} $ is said to be a graded weakly $ J_{gr} $-semiprime submodule of $ \mathcal{D} $ if whenever $ r_{g}\in h(\mathcal{A}), $ $ m_{h}\in h(\mathcal{D}) $ and $ n\in \mathbb{Z} ^{+} $ with $ 0\neq r_{g}^{n}m_{h}\in U $, then $ r_{g}m_{h}\in U+J_{gr}(\mathcal{D}) $, where $ J_{gr}(\mathcal{D}) $ is the graded Jacobson radical of $ \mathcal{D}. $</p></abstract>