We study (1 + 2v)-constacyclic codes over R + vR and their Gray images, where v + v = 0 and R is a finite chain ring with maximal ideal  and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined,...

The minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of C(X). The necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) I is an intersection of O , s, when X is basically disconnected. (iii) I=O , when x X h...

the minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of c(x). the necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) i is an intersection of o , s, when x is basically disconnected. (iii) i=o , when x x ha...

Let U (n) denote the maximal length arithmetic progression in a non-uniform random subset of {0, 1} n , where 1 appears with probability pn. By using dependency graph and Stein-Chen method, we show that U (n) − cn ln n converges in law to an extreme type distribution with ln pn = −2/cn. Similar result holds for W (n) , the maximal length aperiodic arithmetic progression (mod n). An arithmetic p...

Yes, but such a formula is complicated. For example, is there a polynomial f ∈ Z[x] for which f(n) = pn? f(x) = anx n + · · ·+ a1x+ a0 f(a0) = ana0 n + · · ·+ a1a0 + a0 so a0 | f . Suppose q is prime and f(n) = q. Then q | f(n + kq) for each k ∈ Z. So, in particular, we see that if f(m) is prime for each positive integer m, then f is a constant. In particular, f(x) = q for some prime q. The pol...

Let $G$ be a finite group and let $GK(G)$ be the prime graph of $G$. We assume that $ngeqslant 5 $ is an odd number. In this paper, we show that the simple groups $B_n(3)$ and $C_n(3)$ are 2-recognizable by their prime graphs. As consequences of the result, the characterizability of the groups $B_n(3)$ and $C_n(3)$ by their spectra and by the set of orders of maximal abelian subgroups are ...

Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...

It is shown that the parameters of a linear code over Fq of length n, dimension k, minimum weight d and maximum weight m satisfy a certain congruence relation. In the case that q = p is a prime, this leads to the bound m ≤ (n− d)p− e(p− 1), where e ∈ {0, 1, . . . , k− 2} is maximal with the property that ( n− d e ) 6≡ 0 (mod pk−1−e). Thus, if C contains a codeword of weight n then n ≥ d/(p− 1) ...

Let $P$ be a complex polynomial of the form $P(z)=zdisplaystyleprod_{k=1}^{n-1}(z-z_{k})$,where $|z_k|ge 1,1le kle n-1$ then $ P^prime(z)ne 0$. If $|z|

n(n + d) . . . (n + (k − 1)d) = by (1.1) in positive integers n, k ≥ 2, d > 1, b, y, l ≥ 3 with l prime, gcd(n, d) = 1 and P (b) ≤ k. We write d = D1D2 (1.2) where D1 is the maximal divisor of d such that all prime divisors of D1 are congruent to 1 ( mod l). Thus D1 and D2 are relatively prime positive integers such that D2 has no prime divisor which is congruent to 1 (mod l). Shorey [Sh88] pro...