We present the concept of maps Γ- periodi2 on Γ -near-ring S. Our main goal is to research and explore the presence and mapping traits such as h Γ –hom anti-Γ –hom, Γ –α-derivations of Γ -periodi2 on Γ- near-rings.

Let R be a 2-torision free prime ring and ?, ?? Aut(R). Furthermore, G: R×R?R is a symmetric generalized (?, ?)-Biderivation associated with a nonzero (?, ?)-Biderivation D. In this paper some certain identities are presented satisfying by the traces of G and D on an ideal of R which forces R to be commutative

     In this paper, the structure of  and  have  been introduced and studied. We also obtain that a is  of a  if and only if there exists an  on such that . In addition, we obtain  that  of if and only if there is an   on  such that  , where  are subspaces of  with eigenvalues 1 and  −1, respectively. We also find  t that the existence of  on  implies that there exists a compatible  under appropriate condition.

The synthesis and properties of two new series of compounds having 1,3-Oxazepineand 1,3-thiazole rings connected through azo linkage are reported. These compounds weresynthesized by the reaction of phthalic anhydride with Schiff bases. The molecular structuresof these compounds were verified by elemental analysis, FTIR and 1HNMR spectroscopy.The mesomorphic behaviors of these compounds were studied by optical polarizedmicroscopy (OPM) and differential scanning calorimetry (DSC). All compounds of the twoseries show liquid crystalline properties. The influence of the central oxazepine and thiazolerings and the terminal substituents on the type and temperature range of the mesomorphousproperties of these compounds has been elucidated

In this paper we show the nilpotency of nilpotent derivation of simeprime Γ-ring with characteristic 2 must be a power of 2 and we show the nilpotency of a nilpotent derivation of simeprime Γ-ring is either odd or a power of 2 without torsion condition.

Let M be ,-ring and X be ,M-module, Bresar and Vukman studied orthogonal
derivations on semiprime rings. Ashraf and Jamal defined the orthogonal derivations
on -rings M. This research defines and studies the concepts of orthogonal
derivation and orthogonal generalized derivations on ,M -Module X and introduces
the relation between the products of generalized derivations and orthogonality on
,M -module.

     Let S be an inverse semiring, and U be an ideal of S. In this paper, we introduce   the concept of U-S Jordan homomorphism of inverse semirings, and extend the result  of  Herstein on Jordan homomorphisms in inverse semirings.

      In this paper, we introduce the notion of a 2-prime module as a generalization of prime module E over a ring R, where E is said to be prime module if (0) is a prime submodule. We introduced the concept of the 2-prime R-module. Module E is said to be 2-prime if (0) is 2-prime submodule of E. where a proper submodule K of module E is 2-prime submodule if, whenever rR, xE, E, Thus xK or [K: E].

In this paper, we will prove the following theorem, Let R be a ring with 1 having
a reverse derivation d ≠ 0 such that, for each x R, either d(x) = 0 or d(x) is
invertible in R, then R must be one of the following: (i) a division ring D, (ii) D 2 ,
the ring of 2×2 matrices over D, (iii) D[x]/(x ) 2
where char D = 2, d (D) = 0 and
d(x) = 1 + ax for some a in the center Z of D. Furthermore, if 2R ≠ 0 then R = D 2 is
possible if and only if D does not contain all quadratic extensions of Z, the center of
D.