We give an example of a finitely generated commutative semigroup that is not automatic.

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: whenever a direct product \prod _(n \in ℕ) M_n of finitely generated indecomposable modules M_n is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the M_n. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

In this paper, we show that Weyl's theorem holds for operators having the single valued extension property and quasisimilarity preserves Weyl's theorem for these operators under some assumptions for spectral subsets, respectively.

In this paper we consider the integral Volterra operator on the space L^2(0,1). We say that a complex number \lambda is an extended eigenvalue ofV if there exists a nonzero operator X satisfying the equation XV=\lambda VX. We show that the set of extended eigenvalues of V is precisely the interval (0,\infty ) and the corresponding eigenvectors may be chosen to be integral operators as well.

It is proved that the subalgebra of constants of a derivation on a free associative algebra in prime characteristic is free provided that some constraining conditions are satisfied. As a particular case, it follows that the constants of the partial derivatives on a free algebra form a free subalgebra. The main result is also applied in order to provide a simplified proof of a previous result by the author on extensions of tensor rings.

AC-operators generalise normal operators on Hilbert space in the context of well-boundedness. In this paper we study AC-operators T=U+iV, where U and V are commuting well-bounded operators with decomposition of the identity of bounded variation. We also explore some properties of AC-operators by applying the theory of (Foiaş) decomposable operators.

We characterize the pairs of operator spaces that occur as pairs of Morita equivalence bimodules between non-selfadjoint operator algebras in terms of the mutual relation between the spaces. We obtain a characterization of the operator spaces which are completely isometrically isomorphic to imprimitivity bimodules between some strongly Morita equivalent (in the sense of Rieffel) C*-algebras. As corollaries, we give representation results for such operator spaces.

This paper considers the long-time behavior of solutions for the Cauchy problem of a class of second-order nonlinear differential equations: -x''+ f(t,x,x')x'+g(x)=h(t). Under appropriate conditions it is shown that the solutions of the problem possess some dichotomy properties.

In this paper we show that there are no real hypersurfaces in a non-flat complex space form with recurrent Ricci tensor.

We give a method for embedding a large family of partially ordered simple groups of rank one into simple Riesz groups of rank one. In particular, we answer in the affirmative a question of Wehrung, by constructing a torsion-free, simple Riesz group G of rank one containing an interval D\varsubsetneqq G^+ such that 2D=G^+. We sketch some potential applications of this result in the context of monoids of intervals and K-Theory of rings.

If R is a complete discrete valuation ring and M is a reduced, torsion-free R-module of rank \kappa, where \aleph_0 \leq \kappa < 2^ (\aleph_0), we show that M \prop\oplus_(\aleph_0) R \oplus C for some R-module C. As a consequence, it must be the case that M \prop M \oplus (\oplus{_\alpha}R), where \alpha \leq \aleph_0, and {\rm (End)_R}M/\rm (Fin)M has rank at least 2^ (\aleph_0), where Fin M denotes the set of endomorphisms of M with finite rank image.