On Zero Determinant Matrices that are Full

Column-row products have zero determinant over any commutative ring. In this paper we discuss the converse. For domains, we show that this yields a characterization of pre-Schreier rings, and for rings with zero divisors we show that reduced pre-Schreier rings have this property.Finally, for the rings of integers modulo n, we determine the 2x2 matrices which are (or not) full and their numbers.

Two graded rings of Hermitian modular forms

We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in $${\mathbb {Q}}(\sqrt{-7})$$ Q ( - 7 ) and $${\mathbb {Q}}(\sqrt{-11})$$ Q ( - 11 ) . In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms.

Probabilistic and deterministic analogues of the Miller – Rabin algorithm for ideals of rings of integer algebraic elements of finite extensions of the field 

In this paper, we obtained the primality criteria for ideals of rings of integer algebraic elements of finite extensions of the field Q, which are analogues of Miller and Euler’s primality criteria for rings of integers. Also advanced analogues of these criteria were obtained, assuming the extended Riemann hypothesis. Arithmetic and modular operations for ideals of rings of integer algebraic elements of finite extensions of the field Q were elaborated. Using these criteria, the polynomial probabilistic and deterministic algorithms for the primality testing in rings of integer algebraic elements of finite extensions of the field Q were offered.

Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class [Formula: see text] generalizing the Heisenberg group of ([Formula: see text])-unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral groups, which provide formulae for joint distributions of three statistics on such groups.