Homomorphisms of progenerator modules. (English) Zbl 0654.13016

The authors study homomorphisms defined on progenerator modules over a commutative ring R. They define a relation on such homomorphisms, called homotopy, which is coarser than the usual relation of equivalence and show that the set M(R) of homotopy classes forms a commutative monoid under the operation induced by \(\otimes_ R\). The properties of the covariant functor M(-) from the category of commutative rings to the category of commutative monoids are developed. For a given ring R, a classification of homomorphisms is carried out by determining the algebraic structure of M(R) and then giving a representing homomorphism defined on free modules for each class in M(R). A complete description of M(R) is given in the case when R is a Dedekind domain.
Reviewer: T.W.Hungerford


13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13B99 Commutative ring extensions and related topics
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
Full Text: DOI


[1] Bass, H, K-theory and stable algebra, Inst. hautes études sci. publ. math., 22, 5-60, (1964) · Zbl 0248.18025
[2] Curtis, C; Reiner, I, Representation theory of finite groups and associative algebras, (1962), Interscience New York · Zbl 0131.25601
[3] DeMeyer, F; Ingraham, E, Separable algebras over commutative rings, () · Zbl 0215.36602
[4] Hartshorne, R, Algebraic geometry, (1977), Springer-Verlag New York · Zbl 0367.14001
[5] Krull, W, Matrizen, moduln und verallgemeinerte abelsche gruppen im bereich der ganzen algebraischen zahlen, Heidelberger akad. wiss., 2, 13-38, (1932) · JFM 58.0101.02
[6] Levy, L, Almost diagonal matrices over Dedekind domains, Math. Z., 124, 89-99, (1972) · Zbl 0211.36903
[7] Levy, L, Decomposing pairs of modules, Trans. amer. math. soc., 122, 64-80, (1966) · Zbl 0145.04403
[8] McDonald, B.R, Linear algebra over commutative rings, () · Zbl 0466.51018
[9] Steinitz, E; Steinitz, E, Rechteckige systeme and moduln in algebraischen zahlkörpern, II, Math. ann., Math. ann., 72, 297-345, (1912) · JFM 43.0274.01
[10] Zariski, O; Samuel, P, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.