RGNNA Publishing Group.Mathematical Analysis and its Contemporary Applications2716-98983220210601Biﬂatness of Banach algebras modulo an ideal687468283810.30495/maca.2021.1932071.1013ENMohammad AliAbolfathiDepartment of Mathematics, Faculty of Science, Urmia University, Urmia, IranOluwatosin TemitopeMewomoSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South AfricaJournal Article20210531Biﬂatness 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 contenthttps://www.macajournal.com/article_682838_83a8a11a66143db0a472eefc460fd4a6.pdf