Torsion-free modules over Dedekind rings

We obtain necessary and sufficient conditions in order that an arbitrary pure monoendomorphism of a module decomposed into a direct sum of rank 1 torsion-free modules over a Dedekind ring be an automorphism.     

Exchanging torsion modules over Dedekind domains

The author was partially supported by a grant (Project 1419) from the Research Grants Committee of the University of Alabama.     

Dual automorphism-invariant modules over perfect rings

Under study are the dual automorphism-invariant modules and pseudoprojective modules. Some conditions were found under which the dual automorphism-invariant module over a perfect ring is quasiprojective. We also show that if R is a right perfect ring then a pseudoprojective right R-module M is finitely generated if and only if M is a Hopf module.     

Direct products of simple modules over Dedekind domains

Let R be a (commutative) Dedekind domain and let the R-module M be a direct product of simple R-modules. Then any homomorphism from a closed submodule K of M to M can be lifted to M. Received: 9 December 2002     

Tilting modules over almost perfect domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).     

Group rings over Dedekind rings

Seghal posed the following question: IfA andB are rings, doesA[X,X ⁻¹] ℞B[X,X ⁻¹] implyA ℞B. In general the answer to this question is no. In this note we give an affirmative answer in the case thatA andB are Dedekind rings. The author is research assistant at the NFWO.     

Indecomposable modules over right pure semisimple rings

The aim of this paper is to prove the following result. IfA is a right pure semisimple ring, then it satisfies one of the two following statements:

1. For any positive integern, there are at most finitely many indecomposable right modules of lengthn; or
2. There is an infinite number of integersd such that, for eachd, A has infinitely many indecomposable right modules of lengthd.

The result is derived with the aid of ultraproduct-technique.     

On H-supplemented modules over a right perfect ring

In 1971, Koehler [11 Koehler, A. (1971). Quasi-projective and quasi-injective modules. Pac. J. Math. 36(3):713–720.[Crossref], [Web of Science ®] , [Google Scholar]] proved a structure theorem for quasi-projective modules over right perfect rings by using results of Wu–Jans [22 Wu, L. E. T., Jans, J. P. (1967). On quasi-projectives. Illinois J. Math. 11:439–448. [Google Scholar]]. Later Mohamed–Singh [17 Mohamed, S. H., Singh, S. (1977). Generalizations of decomposition theorems known over perfect rings. J. Aust. Math. Soc. Ser. A 24(4):496–510.[Crossref] , [Google Scholar]] studied discrete modules over right perfect rings and gave decomposition theorems for these modules. Moreover, Oshiro [18 Oshiro, K. (1983). Semiperfect modules and quasi-semiperfect modules. Osaka J. Math. 20:337–372.[Web of Science ®] , [Google Scholar]] deeply studied (quasi-)discrete modules over general rings. In this paper, we consider that decomposition theorems for H-supplemented modules with the condition (D₂) or (D₃) over right perfect rings.     

Embedding Witt rings of Dedekind domains

Let W(R) denote Harrison's Witt ring of the commutative ring R. In case R is a field of characteristic ≠ 2, this is the classical Witt ring based on anisotropic quadratic forms. In this note we determine under what conditions W(R) is embedded in W(S) for certain Dedekind domains R ? S. In particular, an answer is given in case R and S are the integers in algebraic number fields K and L, respectively, with (L: K) odd.     

Self-pure-generators over Dedekind domains

We prove that all pure submodules of a finite rank torsion-free module A over a Dedekind domain are A-generated (i.e. A is a self-pure-generator) if and only if A has a rank 1 direct summand B such that $\mathbf{type}(B)$ is the inner type of A. This result is applied to describe the direct products of torsion-free groups of finite rank which are self-pure-generators.     

Motivic cohomology over Dedekind rings

We study properties of Blochs higher Chow groups on smooth varieties over Dedekind rings. We prove the vanishing of for i > n, and the existence of a Gersten resolution for if the residue characteristic is p. We also show that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture an identification for m invertible, and a Gersten resolution with (arbitrary) finite coefficients. Over a complete discrete valuation ring of mixed characteristic (0,p), we construct a map from motivic cohomology to syntomic cohomology, which is a quasi-isomorphism provided the Bloch-Kato conjecture holds.Supported in part by JSPS, NSF Grant. No. 0070850, and the Alfred P.Sloan Foundation