Units and one-sided units in regular rings
HTML articles powered by AMS MathViewer
- by Gertrude Ehrlich
- Trans. Amer. Math. Soc. 216 (1976), 81-90
- DOI: https://doi.org/10.1090/S0002-9947-1976-0387340-0
- PDF | Request permission
Abstract:
A ring R is unit regular if for every $a \in R$, there is a unit $x \in R$ such that $axa = a$, and one-sided unit regular if for every $a \in R$, there is a right or left invertible element $x \in R$ 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 $R = {\text {End}_A}$ 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.References
- Gertrude Ehrlich, Unit-regular rings, Portugal. Math. 27 (1968), 209–212. MR 266962
- László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
- Leonard Gillman and Melvin Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956), 362–365. MR 78979, DOI 10.1090/S0002-9947-1956-0078979-8
- Melvin Henriksen, On a class of regular rings that are elementary divisor rings, Arch. Math. (Basel) 24 (1973), 133–141. MR 379574, DOI 10.1007/BF01228189
- Irving Kaplansky, Fields and rings, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0269449
- L. A. Skornyakov, Complemented modular lattices and regular rings, Oliver & Boyd, Edinburgh-London, 1964. MR 0169799 J. von Neumann, Continuous rings and their arithmetics, Proc. Nat. Acad. Sci. U.S.A. (1936), 707-713. —, On regular rings, Proc. Nat. Acad. Sci. U.S.A. 22 (1936), 707-713.
- Roger Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256. MR 274511, DOI 10.1090/S0002-9947-1971-0274511-2
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 216 (1976), 81-90
- MSC: Primary 16A30
- DOI: https://doi.org/10.1090/S0002-9947-1976-0387340-0
- MathSciNet review: 0387340