In this paper, we will give another class of normal operator which is (K-N)*
quasi-n-normal operator in Hilbert space, and give some properties of this concept
as well as discussion the relation between this class with another class of normal
operators.

According to the wide appearance of Dedekind sums in different applications on different subjects, a new approach for the equivalence of the essential and sufficient condition for ( ) ( ) in and ( ) ( ) in where ( ) Σ(( ))(( )) and the equality of .two .Dedekind sums with their connections is given. The conditions for ( ) ( ) in which is equivalent to ( )– ( ) in were demonstrated with condition that of does not divide . Some applications for the important of Dedekind sums were given.

In this work, the notion of principally quasi- injective semimodule is introduced, discussing the conditions needed to get properties and characterizations similar or related to the case in modules.

      Let  be an -semimodule with endomorphism semiring Ș. The semimodule  is called principally quasi-injective, if every  -homomorphism from any cyclic subsemimodule of  to  can be extended to an endomorphism of .

  In this work we shall introduce the concept of weakly quasi-prime modules and give some properties of this type of modules.

Let R be a commutative ring with identity and let Mbe a unitary R-module. We shall say that a proper submodule N of M is nearly S-primary (for short NS-primary), if whenever , , with  implies that either  or there exists a positive integer n, such that , where  is the Jacobson radical of M. In this paper we give some new results of NS-primary submodule. Moreover some characterizations of these classes of submodules are obtained.

Let R be a ring and let A be a unitary left R-module. A proper submodule H of an R-module A is called 2-absorbing , if rsa∈H, where r,s∈R,a∈A, implies that either ra∈H or sa∈H or rs∈[H:A], and a proper submodule H of an R-module A is called quasi-prime , if rsa∈H, where r,s∈R,a∈A, implies that either ra∈H or sa∈H. This led us to introduce the concept pseudo quasi-2-absorbing submodule, as a generalization of both concepts above, where a proper submodule H of an R-module A is called a pseudo quasi-2-absorbing submodule of A, if whenever rsta∈H,where r,s,t∈R,a∈A, implies that either rsa∈H+soc(A) or sta∈H+soc(A) or rta∈H+soc(A), where soc(A) is socal of an

Let R be a commutative ring with unity and let M be an R-module. In this paper we
study strongly (completely) hollow submodules and quasi-hollow submodules. We investigate
the basic properties of these submodules and the relationships between them. Also we study
the be behavior of these submodules under certain class of modules such as compultiplication,
distributive, multiplication and scalar modules. In part II we shall continue the study of these
submodules.

Let be a commutative ring with unity and let be a submodule of anon zero left R-module , is called semiprime if whenever , implies . In this paper we say that is nearly semiprime, if whenever , implies ( ),(in short ),where ( )is the Jacobson radical of . We give many results of this type of submodules.

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ? W ? M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of ri

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ⊊ W ⊆ M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of rings