In this paper, we study biprojectivity of generalized module extension Banach algebras and second dual of Banach algebras. As a main result, we prove that if I is a contractible closed ideal of A such that A/I is biprojective, then A is biprojective. As a consequence, we give some results on biprojectivity of generalized module extension Banach algebras. Indeed, this is a generalization of results in [A. Ebaddian and A. Jabbari, Biprojectivity and biflatness of amalgamated duplication of Banach algebras, J. Algebra Its Appl. 19(7) (2020) 2050132 (15 pp.)]. Moreover, we show that if A∗∗ is biprojective and A is a two-sided ideal of A∗∗, then A is biprojective. This fact generalizes some known results in [M. S. Moslehian and A. Niknam, Biflatness and biprojectivity of second dual of Banach algebras, Southeast Asian Bull. Math 27(1) (2003) 129–133].

中文翻译：

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Banach代数的广义模扩展和第二对偶的双射性

在本文中，我们研究了广义模扩展Banach代数和Banach代数的第二对偶的双射性。作为主要结果，我们证明如果一世是一个可收缩的封闭理想一种这样一种/一世是双射的，那么一种是双射的。因此，我们给出了关于广义模扩展 Banach 代数的双射性的一些结果。事实上，这是 [A. Ebaddian 和 A. Jabbari，Banach 代数合并重复的双射性和双平面性，J.代数其应用。 19(7) (2020) 2050132 (15 页)]。此外，我们证明如果一种**是双射的并且一种是一个两面的理想一种**,然后一种是双射的。这一事实概括了 [MS Moslehian 和 A. Niknam, Biflatness and biprojectivity of second dual of Banach algebras,东南亚公牛。数学 27(1) (2003) 129–133]。