A denumerable structure is said to be recursive if its universe is a recursive subset of the set of natural numbers and its relations and operations are recursive. A structure is said to be recursively presentable if it is isomorphic to a recursive structure. \textit{L. Feiner} [J. Symb. Logic 35, 365-374 (1970; Zbl 0222.02048)] gave an example of a recursively presentable Boolean algebra \({\mathfrak B}\) with an r.e. ideal I such that \({\mathfrak B}/I\) is not recursively presentable. For \({\mathfrak B}^ a \)Boolean algebra, F(\({\mathfrak B})\) denotes the Fréchet ideal, i.e. the ideal generated by the atoms of \({\mathfrak B}\), and B(\({\mathfrak B})\) denotes the ideal of atomless elements. A sequence of Fréchet ideals is defined as follows: \(F_ 0({\mathfrak B})=\{0\}\), \(F_{\alpha +1}({\mathfrak B})=\{a\in {\mathfrak B}:\) \(a/F_{\alpha}({\mathfrak B})\in F({\mathfrak B}/F_{\alpha}({\mathfrak B}))\}\), and for \(\gamma\) a limit ordinal, \(F_{\gamma}({\mathfrak B})=\cup_{\beta <\gamma}F_{\beta}({\mathfrak B})\). \({\mathfrak B}\) is said to be \(\alpha\)-atomic if \({\mathfrak B}/F_{\beta}({\mathfrak B})\) is atomic for each \(\beta <\alpha.\) The author shows that for each natural number k, there exists a recursively presentable Boolean algebra \({\mathfrak A}\) such that \({\mathfrak A}/B({\mathfrak A})\) is not recursively presentable, and is k-atomic but not \((k+1)\)-atomic. It is further shown that for each atomic countable Boolean algebra \({\mathfrak A}\), there exist \(2^{\aleph_ 0}\) pairwise nonisomorphic countable Boolean algebras \({\mathfrak B}\) with \({\mathfrak B}/B({\mathfrak B})\cong {\mathfrak A}\) if and only if \(F_{\omega}({\mathfrak A})\neq {\mathfrak A}\).