Transitivity and full transitivity for nontorsion modules
Recall that an \(R\)-module \(M\) is (fully) transitive if for all \(x,y\in M\) with \(U_M(x)=U_M(y)\) (\(U_M(x)\leq U_M(y)\)) there exists an \(R\)-automorphism (\(R\)-endomorphism) \(\phi\) of \(M\) such that \(x\phi=y\), where \(U_M(x)\) denotes the height sequence of \(x\) in \(M\). In the first part of the paper the author studies weakened notions of transitivity and full transitivity which are specially suited for use with mixed modules. The second part is devoted to the connection between transitivity and full transitivity for mixed modules. It is shown that these two notions are equivalent for the class of Warfield modules (direct summands of simply presented modules). Another result asserts that a completely decomposable module is fully transitive if and only if its square is transitive. For torsion modules this was shown by the author and \textit{B. Goldsmith} [in Proc. Am. Math. Soc. 126, No. 6, 1605-1610 (1998)]. The author conjectures that all modules obey such a law.
scientific article; zbMATH DE number 1104109
scientific article