let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...

A generic M-ary continuous phase modulation (CPM) scheme, with a modulation index of h = M 1 , can be decomposed into a continuous-phase encoder (CPE) followed by a memoryless modulator (MM), where the CPE is linear over the ring of integers modulo M (M being the code radix). Thus, for a binary CPM scheme, M = 2 while for a quaternary scheme, M = 4. By designing a channel encoder (CE), which is...

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p for all p in a density one subset of the primes. Moreover, the ord...

For every finite abelian group G, there are positive integers n and d such that G is isomorphic to the multiplicative of d-th powers reduced residues modulo n.

In this paper, we investigate the existence problem of perfect linear codes over the ring Z4 of integers modulo 4 with respect to homogenous metric. Mathematics Subject Classification: 94B05; 94B60

Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. $n, k$ and $c$ integers such that $n\ge 1$ $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In this paper, we use principle cross-classification to derive explicit formula for number ${\mathcal N}_{k,f,c}(n)$ solutions $(x_1,...,x_k)$ congruence $x_1+...+x_k\equi...

Let Q(n) denote the number of partitions of an integer n into distinct parts. For positive integers j, the first author and B. Gordon proved that Q(n) is a multiple of 2j for every non-negative integer n outside a set with density zero. Here we show that if i 6≡ 0 (mod 2j), then #{0 ≤ n ≤ X : Q(n) ≡ i (mod 2)} j √ X / logX. In particular, Q(n) lies in every residue class modulo 2j infinitely of...