Lawrence S. Levy

Modules over dedekind-like rings

Abstract All finitely generated modules are described over a class of rings that includes the integral group ring ZGn of the cyclic group of square-free order n, some rings of algebraic integers that are not integrally closed in their field of quotients, and many subrings of Z ⊕ Z ⊕ … ⊕ Z. Special attention is paid to behavior of these modules in direct sums.

Mixed modules over ZG, G cyclic of prime order, and over related Dedekind pullbacks☆

The structure of all finitely generated ZG modules is determined, with emphasis on those modules whose additive group is partly torsion and partly torsion free. The method is to first find all finitely generated R-modules, where R is a pullback of two Dedekind domains over a field; then to specialize to R = ZG, and to other familiar rings.

Matrices and pairs of modules

Abstract It is proved that each matrix over a principal ideal ring is equivalent to some diagonal matrix. Partial results are obtained on the uniqueness of the diagonal form obtained. These results are obtained by specializing some general properties about simultaneous decompositions of a projective module and a homomorphic image of finite (composition) length over any ring. These general results are also specialized to obtain results about matrices and projective modules over hereditary prime rings.

Commutative orders whose lattices are direct sums of ideals

Abstract This paper determines which commutative orders have the property that every finitely generated torsionfree module is isomorphic to a direct sum of ideals.

Representation type of commutative Noetherian rings. III. Global wildness and tameness

Introduction Preliminaries Dedekind-like rings Wildness Structure of a genus Substitute for conductor squares Isomorphism classes in a genus, idele group action Web of class groups Direct sums Finite normalization Appendix A Appendix B Bibliography.

Cancellation and direct summands in dimension 1

Abstract Let Λ be a module-finite algebra over a commutative noetherian ring of Krull dimension 1. We extend Roiters direct-summand theorem to arbitrary finitely generated Λ-modules, obtaining a sharpened form of Serres direct-summand theorem in this setting. We also extend Drozds cancellation theorem to arbitrary finitely generated Λ-modules, obtaining a sharpened form of Basss cancellation theorem in this setting. A corollary is that, over commutative reduced noetherian rings of dimension 1, direct-sum cancellation holds in every genus of finitely generated modules. (This becomes false if the ring has nilpotent elements.) Another corollary is that if direct-sum cancellation holds in the genera of Λ-modules M and N, then it holds in the genus of M ⊕ N. This seems to be new, even for the modules that occur in integral representation theory. The main thrust of this paper is to close the gap between integral representation theory and the rest of module theory by eliminating hypotheses concerning the existence of maximal orders (of “finite normalization,” in the commutative case) and allowing our rings to have nilpotent ideals.

ZGn-modules, Gn cyclic of square-free order n

Abstract In the companion paper ( J. Algebra 93 (1985), 1–116) all finitely generated modules over a class of rings called Dedekind-like are described, with emphasis on the behavior of these modules in direct sums. The present paper begins by showing that the integral group ring ZG n is Dedekind-like. Some properties of ZG n -modules are studied that do not hold for Dedekind-like rings in general. The modules studied do not necessarily have torsion-free abelian groups.

Krull-Schmidt theorems in dimension 1

Let Λ be a semiprime, module-finite algebra over a commutative noetherian ring R of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated Λmodules, and necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated torsionfree Λ-modules (called “Λ-lattices” in integral representation theory, and “maximal Cohen-Macaulay modules” in the dimension-one situation in commutative algebra).

Package deal theorems and splitting orders in dimension 1

Let Λ be a module-finite algebra over a commutative noetherian ring R of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations Λm, at maximal ideals of R, is the family of all localizations Mm of a finitely generated Λ-module M . When R is semilocal we also determine which finitely generated modules over the J(R)-adic completion of Λ are completions of finitely generated Λ-modules. If Λ is an R-order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a “splitting order” of Λ that can replace maximal orders in many situations in which maximal orders do not exist.

Sweeping-similarity of matrices

Abstract This partly expository paper deals with a canonical-form problem for finite sets of matrices. The problem generalizes matrix equivalence, matrix similarity, and simultaneous equivalence of pairs of matrices [( A , B ) → ( PAQ , PBQ )], as well as some more complicated matrix problems that originated in work of Nazarova and Roiter.