Rad-supplemented modules.
Let \(R\) be an associative ring with identity and \(\tau\) a radical for \(R\)-mod, then a submodule \(V\) of a left \(R\)-module \(M\) is said to be a \(\tau\)-supplement (of \(U\)) in \(M\) if there exists a submodule \(U\) of \(M\) such that \(U+V=M\) and \(U\cap V=\tau(V)\). A module \(M\) is called a \(\tau\)-supplemented module if every submodule of \(M\) has a \(\tau\)-supplement in \(M\). Let \(\text{Rad\,}M\) denote the radical of \(M\), then for \(\tau=\text{Rad}\), the above definitions give Rad-supplement submodules of a module and Rad-supplemented modules. Although the paper focuses on Rad-supplemented modules, \(\tau\)-supplemented, for any radical \(\tau\), are examined first. Some properties of \(\tau\)-supplements and \(\tau\)-supplemented modules are found, complementing the known results of \textit{K. Al-Takhman, C. Lomp} and \textit{R. Wisbauer} [Algebra Discrete Math. 2006, No. 3, 1-15 (2006; Zbl 1116.16005)] and those of \textit{J. Clark, C. Lomp, N. Vanaja} and \textit{R. Wisbauer}, [Lifting modules. Supplements and projectivity in module theory. Frontiers in Mathematics. Basel: Birkhäuser (2006; Zbl 1102.16001)]. For example, if \(V\) is a \(\tau\)-supplement in a module \(M\), then it is shown that \(\tau(V)=V\cap\tau(M)\). Thus, for \(V\) a Rad-supplement, we have \(\text{Rad\,}V=V\cap\text{Rad\,}M\). Also, if \(M\) is a \(\tau\)-supplemented module, then \(\text{Rad\,}M\leq\tau(M)\), and \(\text{Rad}(M/P_\tau(M))=\tau(M/P_\tau(M))=\tau(M)/P_\tau(M)\), where \(P_\tau(M)=\sum\{U\leq M\mid\tau(U)=U\}\). Subsequently the following question is studied: When are all left \(R\)-modules \(\tau\)-supplemented? The main result of this section is: For a ring \(R\) with \(P_\tau(R)\leq J(R)\) (the Jacobson radical), the following are equivalent: (i) every left \(R\)-module is \(\tau\)-supplemented, (ii) every free left \(R\)-module is \(\tau\)-supplemented, (iii) the free left \(R\)-module \(F=R^{\mathbb N}\) is \(\tau\)-supplemented, (iv) the quotient ring \(R/P_\tau(R)\) is left perfect and \(\tau(R)=J(R)\). For \(\tau=\text{Rad}\), the question above is investigated further. For any ring \(R\), the following are shown to be equivalent: (i) every left \(R\)-module is Rad-supplemented, (ii) every reduced left \(R\)-module is Rad-supplemented, (iii) every reduced left \(R\)-module is supplemented, (iv) the free left \(R\)-module \(R^{\mathbb N}\) is Rad-supplemented, and (v) \(R/P(R)\) is left perfect, where \(P(R)=P_{\text{Rad}}(R)\). For \(R\) a left duo ring such that \(P(R)=0\), it is shown that \(R\) is a Rad-supplemented left \(R\)-module if and only if \(R\) is semiperfect. Also, the left \(R\)-module \(R\) is Rad-supplemented if and only if \(R/P(R)\) is semiperfect if and only if every finitely generated (free) left \(R\)-module is Rad-supplemented. The final part of the paper deals with the case where \(R\) is a Dedekind domain, the main result stating that an \(R\)-module \(M\) over a Dedekind domain is Rad-supplemented if and only if \(M/P(M)\) is (Rad)-supplemented. This characterization is used to give examples of Rad-supplemented modules that are not supplemented.
scientific article; zbMATH DE number 5849501
scientific article