Let p be an odd prime. We first get some non-existence and structural results on (pn,p,pn,n) relative difference sets with gcd(p,n)= 1 through a group ring approach. We then give a construction of (p(p+ 1),p,p(p+ 1),p+ 1) relative difference sets with p a Mersenne prime.

Let p be a prime number. This paper describes the primitive idempotents and prime spectrum of the crossed Burnside algebra of a finite group over a p-local ring. The main application is a formula for the block idempotents of the p-local Mackey algebra of the group, in terms of the corresponding blocks of the group algebra. Mathematics Subject Classification (2000): 19A22 20C20

In this note, we consider the equiprime and strongly prime radicals in the near-ring LX of Lipschitz functions over a normed vector space X. We prove that the equiprime radical of LX is trivial, and we also obtain upper and lower bounds on its strongly prime radical in terms of the ideal of bounded Lipschitz functions on X. 2000 AMS Classification: 16Y30.

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if charac...

Let $R$ be a commutative ring and $I$ an ideal of $R$. The zero-divisor graph of $R$ with respect to $I$, denoted by $Gamma_I(R)$, is the simple graph whose vertex set is ${x in Rsetminus I mid xy in I$, for some $y in Rsetminus I}$, with two distinct vertices $x$ and $y$ are adjacent if and only if $xy in I$. In this paper, we state a relation between zero-divisor graph of $R$ with respec...

Let G be an abelian group and let R be a commutative G-graded super-ring (briefly, graded super-ring) with unity 1 6= 0. We say that a ∈ h(R), where h(R) is the set of homogeneous elements in R, is weakly prime to a graded superideal I of R if 0 6= r a ∈ I , where r ∈ h(R), then r ∈ I . If ν(I ) is the set of homogeneous elements in R that are not weakly prime to I , then we define I to be weak...

Let R be a ring with an automorphism σ. An ideal I of R is σ-ideal of R if σ(I) = I. A proper ideal P of R is σ-prime ideal of R if P is a σ-ideal of R and for σ-ideals I and J of R, IJ ⊆ P implies that I ⊆ P or J ⊆ P . A proper ideal Q of R is σ-semiprime ideal of Q if Q is a σ-ideal and for a σ-ideal I of R, I2 ⊆ Q implies that I ⊆ Q. The σ-prime radical is defined by the intersection of all ...

We present a (random) polynomial-time algorithm to generate a randomGaussian integer with the uniform distribution among those with norm at most N , along with its prime factorization. The method generalizes to finding a random ideal in the ring of integers of a quadratic number field together with its prime ideal factorization. We also discuss the analogous problem for higher degree number fie...

let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

Let R be an Abelian exchange ring. We prove the following results: 1. RZ2 and RS3 are clean rings. 2. If G is a group of prime order p and p is in the Jacobson radical of R, then RG is clean. 3. If identity in R is a sum of two units and G is a locally finite group, then every element in RG is a sum of two units. 4. For any locally finite group G, RG has stable range one. All rings in this note...