The Axiomatics of Free Group Rings

In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${\mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${\mathbb Z}[F]$.

An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field

Let F be a non-archimedean local field of characteristic 0, and 𝔬 {{\mathfrak{o}}} the ring of integers in F. We give an explicit formula for the Siegel series of a half-integral matrix over 𝔬 {{\mathfrak{o}}} . This formula expresses the Siegel series of a half-integral matrix B explicitly in terms of the Gross–Keating invariant of B and its related invariants.

Notes about symmetric m-adic complexity of generalized cyclotomic sequences of order two with period pq

In this paper, we consider binary generalized cyclotomic sequences with period pq, where p and q are two distinct odd primes. These sequences derive from generalized cyclotomic classes of order two modulo pq. We investigate the generalized binary cyclotomic sequences as the sequences over the ring of integers modulo m for a positive integer m and study m-adic complexity of sequences. We show that they have high symmetric m-adic complexity. Our results generalize well-known statements about 2-adic complexity of these sequences.

A cubic analogue of the Friedlander–Iwaniec spin over primes

In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $$a^2+b^4$$ a 2 + b 4 . To show this they used the Jacobi symbol to define the spin of Gaussian integers, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $${\mathbb {Z}}[\zeta _{12}]={\mathbb {Z}}[\zeta _3,i]$$ Z [ ζ 12 ] = Z [ ζ 3 , i ] by using the cubic residue character on the Eisenstein integers $${\mathbb {Z}}[\zeta _3]$$ Z [ ζ 3 ] . Our main theorem says that the cubic spin is equidistributed along prime ideals of $${\mathbb {Z}}[\zeta _{12}]$$ Z [ ζ 12 ] . The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $$a^2+b^6$$ a 2 + b 6 on the Eisenstein integers.

On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field

Let $${\mathcal {M}}$$ M denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote $$k^{\times }$$ k × , and a uniformizer we denote $$\pi $$ π . In this paper, we consider the map $$T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}$$ T v : M → M defined by $$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$ T v ( x ) = π v ( x ) x - b ( x ) , where b(x) denotes the equivalence class to which $$\frac{\pi ^{v(x)}}{x}$$ π v ( x ) x belongs in $$k^{\times }$$ k × . We show that $$T_v$$ T v preserves Haar measure $$\mu $$ μ on the compact abelian topological group $${\mathcal {M}}$$ M . Let $${\mathcal {B}}$$ B denote the Haar $$\sigma $$ σ -algebra on $${\mathcal {M}}$$ M . We show the natural extension of the dynamical system $$({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)$$ ( M , B , μ , T v ) is Bernoulli and has entropy $$\frac{\#( k)}{\#( k^{\times })}\log (\#( k))$$ # ( k ) # ( k × ) log ( # ( k ) ) . The first of these two properties is used to study the average behaviour of the convergents arising from $$T_v$$ T v . Here for a finite set A its cardinality has been denoted by $$\# (A)$$ # ( A ) . In the case $$K = {\mathbb {Q}}_p$$ K = Q p , i.e. the field of p-adic numbers, the map $$T_v$$ T v reduces to the well-studied continued fraction map due to Schneider.

On (2,3)-generation of matrix groups over the ring of integers, II

Final steps are done in proving that the groups SL ⁡ ( n , Z ) \operatorname {SL}(n,\mathbb {Z}) , GL ⁡ ( n , Z ) \operatorname {GL}(n,\mathbb {Z}) and PGL ⁡ ( n , Z ) \operatorname {PGL}(n,\mathbb {Z}) are ( 2 , 3 ) (2,3) -generated if and only if n ≥ 5 n\ge 5 , and PSL ⁡ ( n , Z ) \operatorname {PSL}(n,\mathbb {Z}) is ( 2 , 3 ) (2,3) -generated if and only if n = 2 n=2 or n ≥ 5 n\ge 5 . In particular, the results cover the remaining cases of n = 8 n=8 , …, 12 12 , and 14 14 .

Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

On the normalized Laplacian spectrum of the comaximal graphs

Let [Formula: see text] be a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be an induced subgraph of [Formula: see text] with nonunit elements of [Formula: see text] as vertices. In this paper, we describe the normalized Laplacian spectrum of [Formula: see text], and we determine it for some values of [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text]. Moreover, we investigate the normalized Laplacian energy and general Randic index of [Formula: see text].