In this paper we introduce the notions of G∗L-module and G∗L-module whichare two proper generalizations of δ-lifting modules. We give some characteriza tions and properties of these modules. We show that a G∗L-module decomposesinto a semisimple submodule M1 and a submodule M2 of M such that every non-zero submodule of M2 contains a non-zero δ-cosingular submodule.

a module m is called epi-retractable if every submodule of m is a homomorphic image of m. dually, a module m is called co-epi-retractable if it contains a copy of each of its factor modules. in special case, a ring r is called co-pli (resp. co-pri) if rr (resp. rr) is co-epi-retractable. it is proved that if r is a left principal right duo ring, then every left ideal of r is an epi-retractable ...

We will start with introducing congruences and investigating modular arithmetic: the set Z/nZ of “integers modulo n” forms a ring. This ring is a field if and only if n is a prime number. A study of the multiplicative structure leads to Fermat’s Little Theorem (for prime n) and to the Euler phi function and Euler’s generalization of Fermat’s Theorem. Another basic tool is the Chinese Remainder ...

Let R be a commutative ring with identity and let M be a torsion free R-module. Several characterizations of distributive modules are investigated. Indeed, among other equivalent conditions, we prove that M is distributive if and only if any primal submodule of M is irreducible, and, if and only if each submodule of M can be represented as an intersection of irreducible isolated components. MSC...

This study was carried out to determine population structure, growth and reproduction properties of barbel. A total of 198 individuals were sampled. Ages of samples were found between I and VI years, fork lengths between 4.3 and 16.6 cm and total weights between 1.2 and 65.8 g. Length-weight relationship was calculated as W=0.0146×L2.934. Munro’s phi prime index was estimated as 1.95, L∞ ...

The technical details of RSA works on the idea that it is easy to generate modulus by multiplying two sufficiently large prime numbers together, but factorizing number back into original extremely difficult. Suppose \(N=p^r q^s\) are modulus, where \(p\) and \(q\) product unknown unbalance primes for \(2 \leq s<r\). paper proves using an approximation \(\phi(N) \approx\) \(N-N^{\frac{r+8-1}{...

Forward We start with the set of natural numbers, N = {1, 2, 3, . . .} equipped with the familiar addition and multiplication and assume that it satisfies the induction axiom. It allows us to establish division with a residue and the Euclid’s algorithm that computes the greatest commond divisor of two natural numbers. It also leads to a proof of the fundamental theorem of arithmetic: Every natu...

in this paper we investigate decompositions of submodules in modules over a proufer domain into intersections of quasi-primary and classical quasi-primary submodules. in particular, existence and uniqueness of quasi-primary decompositions in modules over a proufer domain of finite character are proved. proufer domain; primary submodule; quasi-primary submodule; classical quasi-primary; decomposi...

The purpose of this paper is to introduce some new classes of rings and modules that are closely related to the classes of almost Dedekind domains and almost Dedekind modules. We introduce the concepts of $\phi$-almost Dedekind rings and $\Phi$-almost Dedekind modules and study some properties of this classes. In this paper we get some equivalent conditions for $\phi$-almost Dedekind rings and ...

Let $p \geq 2$ be a prime, and $\mathbb{F}_p$ the field with $p$ elements. Extending result of Seidel for $p=2,$ we construct an isomorphism between Floer cohomology exact or Hamiltonian symplectomorphism $\phi,$ coefficients, $\mathbb{Z}/p \mathbb{Z}$-equivariant Tate its $p$-th power $\phi^p.$ The construction involves Kaledin-type quasi-Frobenius map, as well pants product: equivariant opera...