We investigated maximal Prym varieties on  nite  elds by attaining their upper bounds on the number of rational points. This concept gave us a motivation for de ning a generalized de ni-tion of maximal curves i. e., maximal MORPHISMs. By MAGMA, we give some non-trivial examples of maximal MORPHISMs that results in non-trivial examples of maximal Prym varieties.

In this paper, we generalize the construction of a pullback diagram in the framework of Hilbert MODULEs over locally C*-algebras and we explore some conditions under which diagrams of Hilbert MODULEs over the corresponding C*-algebras are pullback.

In this paper we defined the concept of MODULE amenability of Banach algebras and MODULE connes amenability of MODULE dual Banach algebras. Also we assert the concept of MODULE Arens regularity that is different with [1] and investigate the relation between MODULE amenability of Banach algebras and connes MODULE amenability of MODULE second dual Banach algebras. In the following we study the relation between MODULE amenability, weak MODULE amenability and MODULE approximate amenability of Banach algebra. The notation of amenability of Banach algebras was introduced by B. Johnson in [7]. A Banach algebra A is amenable if every bounded derivation from A into any dual Banach A-biMODULE is inner, equivalently if H(A; X) = 0 for any Banach A-biMODULE X, where H(A; X) is the first Hochschild co-homology group of A with coefficient in X. Also, a Banach algebra A is weakly amenable if H(A; A) = 0. Bade, Curtis and Dales introduced the notion of weak amenability on Banach algebras in [4]. They considered this concept only for commutative Banach algebras. After that Johnson defined the weak amenability for arbitrary Banach algebras.

Let A be a Banach algebra and E be a Banach A -biMODULE then S=A ÅE, the l1-direct sum of A and E becomes a MODULE extension Banach algebra when equipped with the algebras product (a, x): (a', x')=(aa', a.x'+x.a'). In this paper, we investigate D-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the MODULE extension Banach algebra S=l1 (ES) Å l1 (S) is D-amenable as a l1 (ES) -MODULE if and only if l1 (ES) is amenable as Banach algebra.