In this paper, the Khintchine-type theorems of Beresnevich (Acta Arith. 90 (1999), 97) and Bernik (Acta Arith. 53 (1989), 17) for polynomials are generalised to incorporate a natural restriction on derivatives. This represents the first attempt to solve a problem posed by Bernik, Kleinbock and Margulis (Int. Math. Res. Notices2001(9) (2001), 453). More specifically, the main result provides a probabilistic criterion for the solvability of the system of inequalities |P(x)| < Ψ1(H) and |P′(x)| < Ψ2(H) in integral polynomials P of degree ≤ n and height H, where Ψ1 and Ψ2 are fairly general error functions. The proof builds upon Sprindzuk's method of essential and inessential domains and the recent ideas of Beresnevich, Bernik and Götze (Compositio Math. 146 (2010), 1165) concerning the distribution of algebraic numbers.

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.

Let R = ⊕i ∈ ℤRi be a ℤ-graded ring and M = ⊕i ∈ ℤMi be a graded R-module. Providing some results on graded multiplication modules, some equivalent conditions for which a finitely generated graded R-module to be graded multiplication will be given. We define generalised graded multiplication module and determine some of its certain graded prime submodules. It will be shown that any graded submodule of a finitely generated generalised graded multiplication R-module M has a kind of primary decomposition. Using this, we give a characterisation of graded primary submodules of M. These lead to a kind of characterisation of finitely generated generalised graded multiplication modules.

It is shown that the well-known characterizations of when a commutative ring R has Noetherian spectrum carry over to characterizations of when the set Spec(M) of prime submodules of a finitely generated module M is Noetherian. The symmetric algebra SR(M) of M is used to show that the Noetherian property of Spec(R), and some related properties, pass from the ring R to the finitely generated R-modules.

In this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke–Santaló type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

In this paper, we discuss a connection between (−1, −1)-Freudenthal–Kantor triple systems, anti-structurable algebras, quasi anti-flexible algebras and give examples of such structures. The paper provides the correspondence and characterization of a bilinear product corresponding a triple product.