Biflatness of Banach algebras modulo an ideal

Biflatness of Banach algebras is one of the important topics in the study of cohomological properties of Banach algebras. This concept has a close relationship with the amenability of Banach algebras. In this paper, we introduce a new notion namely biflatness of Banach algebras modulo closed ideals. Moreover, we define the concept of virtual diagonal modulo ideals for investigating biflatness of Banach algebras modulo closed ideals. We show that biflatness of a Banach algebra A modulo I is equivalent to the existence of I -virtual diagonal modulo ideal I. By this result, we show that amenability of A/I implies biflatness of A modulo I. Moreover, we investigate the relationship of biflatness of the Banach algebra A modulo I with the biflatness of A/I . Finally, biflatness of Banach algebras modulo closed ideals is weaker than biprojectivity of them modulo closed ideals and provide examples to better understand the content