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

This qualitative study was conducted on eight types of commercial baking yeast which available in local markets to estimate their fermentation activity as affecting the Bread industry and the impact of the salt added to DoughLeavening, The results showed a great variation in the fermentation capacity of yeast samples (their role in swelling the dough), most notably the sample value Y3 and least sample Y7 and reached 80% and 20% respectively, and the value of Leavening by using the two types of yeast with addition of three levels of salt (0 , 1 and 2%) have 20.0 , 19.7 and 15.7 of the sample Y3, compared with 10.5 , 10.3 and 8.8 of the sample Y7 for each of the levels of salt respectively, reflect

We study in this paper the composition operator that is induced by ?(z) = sz + t. We give a characterization of the adjoint of composiotion operators generated by self-maps of the unit ball of form ?(z) = sz + t for which |s|?1, |t|<1 and |s|+|t|?1. In fact we prove that the adjoint is a product of toeplitz operators and composition operator. Also, we have studied the compactness of C? and give some other partial results.

ABSTRACT : Bacillus cereus and Pseudomonas aeruginosa is the ability to produce a wide antimicrobial active compounds (Bacillin and S-Pyocin) against pathogenic microorganism. In vitro assay with the antagonists of both crude bacteriocin and partial by precipitation 75% ammonium sulfate showed that the effectively inhibited growth of the following (Candida kefyer and Fusarium spp) and Propionibacterium acnes. The results showed the inhibition zone of reached Bacillin (9-13 mm), while Pyocin (13 - 16mm) in solid medium.

Convergence prop erties of Jackson polynomials have been considered by Zugmund
[1,ch.X] in (1959) and J.Szbados [2], (p =ï‚¥) while in (1983) V.A.Popov and J.Szabados [3]
(1 ï‚£p ï‚£ ï‚¥) have proved a direct inequality for Jackson polynomials in L
p-sp ace of 2-periodic bounded Riemann integrable functions (f R) in terms of some modulus of
continuity .
In 1991 S.K.Jassim proved direct and inverse inequality for Jackson polynomials in
locally global norms (L
,p) of 2-p eriodic bounded measurable functions (f L) in terms of
suitable Peetre K-functional [4].
Now the aim of our paper is to proved direct and inverse inequalities for Jackson
polynomials

     The -s-extending modules will be purpose of this paper, a module M  is -s-extending if each submodule in M is essential in submodule has a supplement that is direct summand. Initially, we give relation between this concept with weakly supplement extending modules and -supplemented modules. In fact, we gives the following implications:

Lifting modules   -supplemented modules   -s-extending modules  weakly supplement extending modules.

It is also we give examples show that, the converse of this result is not true. Moreover, we study when the converse of this result is true.

Necessary and sufficient conditions for the operator equation I AXAX n*, to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.

Protecting information sent through insecure internet channels is a significant challenge facing researchers. In this paper, we present a novel method for image data encryption that combines chaotic maps with linear feedback shift registers in two stages. In the first stage, the image is divided into two parts. Then, the locations of the pixels of each part are redistributed through the random numbers key, which is generated using linear feedback shift registers. The second stage includes segmenting the image into the three primary colors red, green, and blue (RGB); then, the data for each color is encrypted through one of three keys that are generated using three-dimensional chaotic maps. Many statistical tests (entropy, peak signa