[en] L. Schwarz (1954) demonstrated the impossibility to define associative multiplication in distributions space. It should be noted that Physcicists often determine and use product of two generalized functions processing from intuitive considerations. Sometimes this leads to erroneous results (Egorov, 1990). In 1984 French mathematician J. F. Colombean studied some algebras of the objects referred to as 'New Generalized Functions' (Colombean, 1985). Ya. V. Egorov has in turn developed a simpler. as compared to Colombeau's theory of generalized functions and defined its applicability to nonlinear differential equations with partial derivatives (Egorov, 1990). The works of A.B. Antonevitch and Ya. V. Radyno give a general construction method for the algebras of new generalized functions and provided examples of its applications (Radyno, 1991). In 1992, and by the general method in (Ibid), we constructed the algebra L(S(R)such that S'(R) / L(S(R) (Radyno et al., 1992). And in 1997 we defined the algebras L(D (R)), and L(Z(R) in such away that the spaces L(Z(R)), and L(D(R)) are included in the algebra L(S(R)) (Ramadan, 1998). And we proved the exact results: S' (R)/ L(Z(R)) / L(S(R)). In this paper we construct the new algebra Lα ₍X), and in particular we define the algebra L_α (S_α α⁽R)). Also we study its properties, and define and study the linear operations A*_α: L_α(X) →L_α (X). (authors)