Let R be a commutative ring with identity and M an R–module. If M is either locally cyclic projective or faithful multiplication then M is locally either zero or isomorphic to R. We investigate locally cyclic projective modules and the properties they have in common with faithful multiplication modules. Our main tool is the trace ideal. We see that the module structure of a locally cyclic proje...

Let $R$ be a commutative ring with identity and $M$ be an$R$-module. Let $FSpec(M)$ denotes the collection of all prime fuzzysubmodules of $M$. In this regards some basic properties of Zariskitopology on $FSpec(M)$ are investigated. In particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$FSpec(M)$ is a $T_0-$spa...

In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for q...

Let A be an abelian variety defined over a non-prime finite field Fq that has embedding degree k with respect to a subgroup of prime order r. In this paper we give explicit conditions on q, k, and r that imply that the minimal embedding field of A with respect to r is Fqk . When these conditions hold, the embedding degree k is a good measure of the security level of a pairing-based cryptosystem...

In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for q...

Let F be a class of finite groups, p a prime and π a set of some primes. A finite group G is called a p-quasi-F-group (respectively, by π-quasi-F-group) provided that for every F-eccentric G-chief factor H/K of order divisible by p (respectively, by at least one prime in π), the automorphisms of H/K induced by all elements of G are inner. In this paper, we obtain the characterizations of p-quas...