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Strongly $\pi $-regular rings
have stable range one


Author: Pere Ara
Journal: Proc. Amer. Math. Soc. 124 (1996), 3293-3298
MSC (1991): Primary 16E50, 16U50, 16E20
DOI: https://doi.org/10.1090/S0002-9939-96-03473-9
MathSciNet review: 1343679
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Abstract: A ring $R$ is said to be strongly $\pi $-regular if for every $a\in R$ there exist a positive integer $n$ and $b\in R$ such that $a^{n}=a^{n+1}b$. For example, all algebraic algebras over a field are strongly $\pi $-regular. We prove that every strongly $\pi $-regular ring has stable range one. The stable range one condition is especially interesting because of Evans' Theorem, which states that a module $M$ cancels from direct sums whenever $\text {End}_{R} (M)$ has stable range one. As a consequence of our main result and Evans' Theorem, modules satisfying Fitting's Lemma cancel from direct sums.


References [Enhancements On Off] (What's this?)

  • 1. P. Ara, K.R. Goodearl, K. O'Meara and E. Pardo, Separative cancellation for projective modules over exchange rings, Preprint.
  • 2. E.P. Armendariz, J.W. Fisher and R.L. Snider, On injective and surjective endomorphisms of finitely generated modules, Communications in Algebra 6(7) (1978), 659-672. MR 57:9754
  • 3. G. Azumaya, Strongly $\pi $-regular rings, J. Fac. Sci. Hokkaido Univ. 13 (1954), 34-39. MR 16:788
  • 4. W.D. Burgess and P. Menal, On strongly $\pi $-regular rings and homomorphisms into them, Communications in Algebra 16(8) (1988), 1701-1725. MR 89f:16015
  • 5. V.P. Camillo and Hua-Ping Yu, Stable range one for rings with many idempotents, Trans. Amer. Math. Soc. 347 (1995), 3141--3147. CMP 95:12
  • 6. R. Camps and P. Menal, Power cancellation for artinian modules, Communications in Algebra 19(7) (1991), 2081-2095. MR 92m:16006
  • 7. P. Crawley and B. Jónsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797-855. MR 30:49
  • 8. M.F. Dischinger, Sur les anneaux fortement $\pi $-reguliers, C.R. Acad. Sci. Paris, Ser. A 283 (1976), 571-573. MR 54:10321
  • 9. E.G. Evans, Krull-Schmidt and cancellation over local rings, Pacific J. Math 46 (1973), 115-121. MR 48:2170
  • 10. K.R. Goodearl, Surjective endomorphisms of finitely generated modules, Communications in Algebra 15(3) (1987), 589-609. MR 88d:16010
  • 11. K.R. Goodearl and P. Menal, Stable range one for rings with many units, J. Pure Applied Algebra 54 (1988), 261-287. MR 89h:16011
  • 12. I. Kaplansky, Topological representations of algebras II, Trans. Amer. Math. Soc. 68 (1950), 62-75. MR 11:317
  • 13. P. Menal, On $\pi $-regular rings whose primitive factor rings are artinian, J. Pure Applied Algebra 20 (1981), 71-78. MR 81k:16015
  • 14. P. Menal, Cancellation modules over regular rings, in ``Ring Theory, Proceedings, Granada 1986", J.L. Bueso, P. Jara and B. Torrecillas (eds.), LNM 1328, Springer-Verlag, 1988, pp. 187-209. MR 89i:16010
  • 15. W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. MR 55:12757
  • 16. J. Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (1986), 437-453. MR 88e:16038
  • 17. L.N. Vaserstein, Stable rank of rings and dimensionality of topological spaces, Funct. Ana. Appl. 5 (1971), 102-110. MR 44:1701
  • 18. L.N. Vaserstein, Bass's first stable range condition, J. Pure Applied Algebra 34 (1984), 319-330. MR 86c:18009
  • 19. R.B. Warfield, Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31-36. MR 48:11218
  • 20. Hua-Ping Yu, Stable range one for exchange rings, J. Pure Applied Algebra 98 (1995), 105-109.
  • 21. Hua-Ping Yu, On strongly pi-regular rings of stable range one, Bull. Austral. Math. Soc. 51 (1995), 433--437. CMP 95:12

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Additional Information

Pere Ara
Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Email: para@mat.uab.es

DOI: https://doi.org/10.1090/S0002-9939-96-03473-9
Keywords: Strongly $\pi $-regular ring, stable range one, exchange ring, Fitting's Lemma
Received by editor(s): April 28, 1995
Additional Notes: The author was partially supported by DGYCIT grant PB92-0586 and the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1996 American Mathematical Society

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