On Steinitz properties

We perform an in-depth study of several different cancellation properties for modules. Among those we consider are (half) (weak) cancellation modules, restricted cancellation modules, and (half) join principal modules. We also investigate which commutative rings have every nonzero (finitely generated) ideal (respectively, module) satisfying some cancellation property.

Module cancellation properties

We introduce a construction adding low-dimensional cells to a space that satisfies certain low-dimensional conditions; it preserves high-dimensional homology with appropriate coefficients. This includes as special cases Quillen’s plus construction, Bousfield’s integral homology localization, the existence of Moore spaces M(G, 1) and Bousfield and Kan’s partial k-completion of spaces. We also use it to generalize counterexamples to the zero-in-the-spectrum conjecture found by Farber and Weinberger, and by Higson, Roe and Schick.

/pdf/a-unified-approach-to-the-plus-construction-bousfield-37jbede50w.pdf

A unified approach to the plus-construction, bousfield localization, moore spaces and zero-in-the-spectrum examples

Let R be a ring, MR be a right R-module, n be a positive integer and S = End(MR) be the endomorphism ring of MR. MR is called a strongly C2 module if $M^m_R$ is C2 for every positive integer m. MR is called an n-C2 module if the annihilator rM(K) ≠ 0 for any n-generated proper left ideal K of S. We prove that MR is strongly C2 if and only if M is n-C2 for every positive integer n, if and only if the annihilator rM(K) is not zero for every finitely generated proper left ideal K of S, and then we get some characterizations of right n-C2 rings and strongly right C2 rings. Also we obtain some dual statements of n-D2 module and strongly D2 module.

On strongly c2 modules and d2 modules

We introduce a construction adding low-dimensional cells to a space that satisfies certain low-dimensional conditions; it preserves high-dimensional homology with appropriate coefficients. This includes as special cases Quillen's plus construction, Bousfield's integral homology localization, the existence of Moore spaces M(G,1) and Bousfield and Kan's partial k-completion of spaces. We also use it to generalize counterexamples to the zero-in-the-spectrum conjecture found by Farber and Weinberger, and by Higson, Roe and Schick.

A unified approach to the plus-construction, Bousfield localization, Moore spaces and zero-in-the-spectrum examples

A ring R (associative with an identity) is called a right Steinitz ring if any linearly independent subset of a free right i?-module F can be extended to a basis of F. Steinitz rings have been investigated in a recent paper of Chwe and Neggers. In this note it is shown that the right Steinitz rings are exactly the right perfect, local rings. 1. Preliminaries. R denotes an associative ring with an identity, and all modules are unitary right i?-modules. Obviously R is a right Steinitz ring iff, for any free right A-module F, and any free sub- module U of F, F/ U is again free (and U a direct summand of F). Every projective i?-module P over a right Steinitz ring R is free: Let P be a direct summand of a free module F of infinite rank, then F=P ®F by a standard argument. Therefore P=P®F/F is free. Conse- quently R is a right Steinitz ring iff any right i?-module of finite homological dimension is free.

/pdf/a-homological-characterization-of-steinitz-rings-5xqfmwdf6o.pdf

A homological characterization of steinitz rings

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A homological characterization of steinitz rings