Analogues of IBN and related properties for modules.
The author introduces and studies module-theoretic versions of several ring properties, namely, invariant basis number, rank condition, stable finiteness and strong rank condition. The corresponding module properties are called, respectively, unique exponent property, exponent condition, stable finiteness and strong exponent condition. The first three of them hold for a module \(M\) if and only if the corresponding ring property holds for the ring \(\text{End}(M)\) but, while the strong rank condition for \(\text{End}(M)\) implies that \(M\) satisfies the strong exponent condition, the converse is not true in general as shown by an example given at the end of the paper. The author studies the effect on each of these properties of standard module-theoretic operations and also the behavior of the discussed properties under change of rings or passing to modules of polynomials or power series. The paper is well written and contains several examples that illustrate the theory.
scientific article; zbMATH DE number 5355121
scientific article