AB-5* and linear compactness
In this interesting paper, the authors study AB5*-modules and linearly compact modules. The notion of linearly compact modules can be found in Section 3 of the reviewer's book ``Rings with Morita duality'' [Lect. Notes Math. 1523 (1992; Zbl 0790.16009)]. A module \(M\) is AB5* if given a submodule \(L\) of \(M\) and a filter basis \(\{L_i\}_{i\in I}\) of submodules of \(M\) (i.e., for each pair \(i,j\in I\) there is a \(k\) such that \(L_k\subseteq L_i\cap L_j\)) then \(L+\bigcap_{i\in I}L_i=\bigcap_{i\in I}(L+L_i)\). It is known that every linearly compact module is AB5* but the converse is false. The authors use a result due to Vamos (see Theorem 3.1) to prove that \(R\) is left linearly compact whenever the \(n\) by \(n\) matrix ring (\(n>1\)) is left AB5*. The reviewer [Can. Math. Bull. 39, No. 2, 250-256 (1996; Zbl 0858.16005)]\ proved that the power series ring \(Rx\) is left linearly compact if and only if \(R\) is left Noetherian and left linearly compact. In this paper, the authors prove that if \(Rx\) is left AB5* then \(R\) is left Noetherian and left AB5*, but the converse is false even in the commutative case. They also investigate the local commutative Noetherian AB5*-rings. Theorem 4.2, Proposition 4.4, and Theorem 4.9 characterize these rings while Proposition 4.10 is a powerful tool for an AB5*-test.
scientific article; zbMATH DE number 1139770
scientific article