Let \( R \) be a commutative Noetherian ring, \( \mathfrak {a}\) an ideal of \( R \) , and \( M \) a finitely generated \( R \) -module. We consider the idealization \( R \ltimes M \) of \( M \) over \( R \) . In Ahmadi and Rahimi (J Algebra Appl 23(12):2450209, 2024) and Taniguchi et al. (J Commut Algebra 10(2):295–304, 2018), the authors study how certain algebraic properties of \( R \ltimes M \) relate to similar properties of \( R \) and \( M \) , and vice versa. Building on their work, we provide a description of quasi-Buchsbaumness, Buchsbaumness, and surjective Buchsbaumness for \( R \ltimes M \) with respect to \( \mathfrak {a}\ltimes M \) where \(\mathfrak {a}\) is an ideal of R. In particular, when \( R \) is a local ring, we describe these properties for the local ring \( R \ltimes M \) .
