Infinite Abelian groups

Residually small varieties

On direct products of modules

A ring R is unit regular if for every a E R, there is a unit x E R such that axa = a, and one-sided unit regular if for every a E R, there is a right or left invertible element x E R such that axa = a. In this paper, unit regularity and one-sided unit regularity are characterized within the lattice of principal right ideals of a regular ring R (Theorem 3). If M is an A-module and R = EndA M is a regular ring, then R is unit regular if and only if complements of isomorphic summands of M are isomorphic, and R is one-sided unit regular if and only if complements of isomorphic summands of M are comparable with respect to the relation "is isomorphic to a submodule of" (Theorem 2). A class of modules is given for whose endomorphism rings it is the case that regularity in conjunction with von Neumann finiteness is equivalent to unit regularity. This class includes all abelian torsion groups and all nonreduced abelian groups with regular endomorphism rings. In [1], a ring R with identity was defined to be unit regular if for every a E R there is a unit x E R such that axa = a. The class of all unit regular rings includes [1] all semisimple Artinian rings, all continuous von Neumann rings [7], all strongly regular rings (in particular, all commutative regular rings [3] ). Using the characterization (cf. [6, p. 117] ) of regular group rings as the group rings of locally finite groups, it is easy to show that all regular group rings are unit regular. Unit regular rings are von Neumann finite [4, Proposition 1] and are elementary divisor rings [4, Theorem 3]. Every element of a unit regular ring in which 2 is a unit is equal to the sum of two units [1, Theorem 6]. An example of a regular ring which is not unit regular is the endomorphism ring of an infinite dimensional vector space [1]. In this paper, we define one-sided unit regularity and characterize both unit regularity and one-sided unit regularity within the lattice of principal right ideals of the ring (Theorem 3). For an A-module M with regular endomorphism ring R we prove that R is unit regular if and only if any two isomorphic complemented Presented to the Society, March 19, 1974; received by the editors April 4, 1974 and, in revised form, September 9, 1974. AMS (MOS) subject classifications (1970). Primary 16A30; Secondary 16A42, 16A48, 16A64, 20K30.

/pdf/units-and-one-sided-units-in-regular-rings-glc4s6f8lp.pdf

Units and one-sided units in regular rings

This paper is a contribution to the study of Borel equivalence relations on standard
Borel spaces (i.e., Polish spaces equipped with their Borel structure). In
mathematics one often deals with problems of classification of objects up to some
notion of equivalence by invariants. Frequently these objects can be viewed as elements
of a standard Borel space X and the equivalence turns out to be a Borel
equivalence relation E on X. A complete classification of X up to E consists of
finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y).
For this to be of any interest both I and c must be explicit or definable and as simple
and concrete as possible. The theory of Borel equivalence relations studies the
set-theoretic nature of possible invariants and develops a mathematical framework
for measuring the complexity of such classification problems.

/pdf/linear-algebraic-groups-and-countable-borel-equivalence-4kr3yvrf10.pdf

Linear algebraic groups and countable Borel equivalence relations

We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal /C to ~IC+I. Hence there are unboundedly many almost free nonfree Abelian groups below the first cardinal fixed point. We give a sufficient condition for \"/c free implies free\", and then we show, assuming the consistency of infinitely many supercompacts, that one can have a model of ZFC+G.C.H. in which ~w2+1 free implies ~w2+2 free. Similar construction yields a model in which ~\" free implies free for /C the first cardinal fixed point (namely, the first cardinal a satisfying a = ~a) . The absolute results about the existence of almost free nonfree groups require only minimal knowledge of set theory. Also, no knowledge of metamathematics is required for reading the section on the combinatorial principle used to show that almost free implies free. The consistency of the combinatorial principle requires acquaintance with forcing techniques. INSTITUTE OF MATHEMATICS, HEBREW UNIVERSITY, JERUSALEM, ISRAEL License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

/pdf/when-does-almost-free-imply-free-for-groups-transversals-etc-qhodd8k0gj.pdf

Infinite Abelian groups

Citations

Residually small varieties

On direct products of modules

Units and one-sided units in regular rings

Linear algebraic groups and countable Borel equivalence relations

When does almost free imply free? (For groups, transversals, etc.)

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