Algebras and Representation Theory

, Volume 15, Issue 6, pp 1049–1079 | Cite as

Strongly Semihereditary Rings and Rings with Dimension



The existence of a well-behaved dimension of a finite von Neumann algebra (see Lück, J Reine Angew Math 495:135–162, 1998) has lead to the study of such a dimension of finite Baer *-rings (see Vaš, J Algebra 289(2):614–639, 2005) that satisfy certain *-ring axioms (used in Berberian, 1972). This dimension is closely related to the equivalence relation \( {\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}\) on projections defined by \(p{\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}} q\) iff p = xx* and q = x*x for some x. However, the equivalence \({\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}}\) on projections (or, in general, idempotents) defined by \(p{\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}} q\) iff p = xy and q = yx for some x and y, can also be relevant. There were attempts to unify the two approaches (see Berberian, preprint, 1988)). In this work, our agenda is three-fold: (1) We study assumptions on a ring with involution that guarantee the existence of a well-behaved dimension defined for any general equivalence relation on projections ~. (2) By interpreting ~ as \({\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}},\) we prove the existence of a well-behaved dimension of strongly semihereditary *-rings with positive definite involution. This class is wider than the class of finite Baer *-rings with dimension considered in the past: it includes some non Rickart *-rings. Moreover, none of the *-ring axioms from Berberian (1972) and Vaš (J Algebra 289(2):614–639, 2005) are assumed. (3) As the first corollary of (2), we obtain dimension of noetherian Leavitt path algebras over positive definite fields. Secondly, we obtain dimension of a Baer *-ring R satisfying the first seven axioms from Vaš (J Algebra 289(2):614–639, 2005) (in particular, dimension of finite AW*-algebras). Assuming the eight axiom as well, R has dimension for \({\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}\) also and the two dimensions coincide. While establishing (2), we obtain some additional results for a right strongly semihereditary ring R: we prove that every finitely generated R-module M splits as a direct sum of a finitely generated projective module and a singular module; we describe right strongly semihereditary rings in terms of relations between their maximal and total rings of quotients; and we characterize extending Leavitt path algebras over finite graphs.


Dimension Rings of quotients Semihereditary Involution Baer Regular 

Mathematics Subject Classification (2010)

16W99 16S99 16S90 16W10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra 293(2), 319–334 (2005)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Abrams, G., Aranda Pino, G., Siles Molina, M.: Locally finite Leavitt path algebras. Isr. J. Math. 165, 329–348 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Abrams, G., Tomforde, M.: Isomorphism and Morita equivalence of graph algebras. Trans. Am. Math. Soc. 363, 3733–3767 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ara, P., Brustenga, M.: The regular algebra of a quiver. J. Algebra 309, 207–235 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ara, P., Menal, P.: On regular rings with involution. Arch. Math. Basel 42(2), 126–130 (1984)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ara, P., Moreno, M.A., Pardo, E.: Nonstable K-theory for graph algebras. Algebr. Represent. Theory 10, 157–178 (2007)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Aranda Pino, G., Rangaswamy, K.L., Vaš, L.: *-regular Leavitt path algebra of arbitrary graphs. Acta Math. Sin. (Eng. Ser.) (to appear, 2011)Google Scholar
  8. 8.
    Aranda Pino, G., Vaš, L.: Noetherian Leavitt path algebras and their regular algebras. (preprint, 2011)Google Scholar
  9. 9.
    Berberian, S.K.: Baer *-rings. Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag, New York (1972)Google Scholar
  10. 10.
    Berberian, S.K.: Baer rings and Baer *-rings. (preprint, 1988) Available at
  11. 11.
    Dung, N.V., Huynh, D.V., Smith, P.F., Wisbauer, R.: Extending Modules. Pitman, London (1994)MATHGoogle Scholar
  12. 12.
    Evans, M.W.: A class of semihereditary rings. Rings, modules and radicals (Hobart, 1987), pp. 51–60, Pitman Res. Notes Math. Ser., vol. 204. Longman Sci. Tech., Harlow (1989)Google Scholar
  13. 13.
    Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger, Malabar, FL (1991)Google Scholar
  14. 14.
    Goodearl, K.R.: Embedding non-singular modules in free modules. J. Pure Appl. Algebra 1, 275–279 (1971)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Huynh, D.V., Rizvi, S.T., Yousif, M.F.: Rings whose finitely generated modules are extending. J. Pure Appl. Algebra 111, 325–328 (1996)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kaplansky, I.: Rings of Operators. Benjamin, New York (1968)MATHGoogle Scholar
  17. 17.
    Lam, T.Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics 189, Springer-Verlag, New York (1999)MATHCrossRefGoogle Scholar
  18. 18.
    Lück, W.: L 2-invariants: Theory and Applications to Geometry and K-theory. Ergebnisse der Mathematik und ihrer Grebzgebiete, Folge 3, 44, Springer-Verlag, Berlin (2002)Google Scholar
  19. 19.
    Lück, W.: Dimension theory of arbitrary modules over finite von Neumann algebras and L 2-Betti numbers I: Foundations. J. Reine Angew. Math. 495, 135–162 (1998)MathSciNetMATHGoogle Scholar
  20. 20.
    Maeda, S., Holland, S.S. Jr.: Equivalence of projections in Baer *-rings. J. Algebra 39, 150–159 (1976)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ortega, E.: Two-sided localization of bimodules. Commun. Algebra 36(5), 1911–1926 (2008)MATHCrossRefGoogle Scholar
  22. 22.
    Osofsky, B.L., Smith, P.F.: Cyclic modules whose quotients have all complement submodules direct summands. J. Algebra 139, 342–354 (1991)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Pyle, E.S.: The regular ring and the maximal ring of quotients of a finite Baer *-ring. Trans. Am. Math. Soc. 203, 201–213 (1975)MathSciNetMATHGoogle Scholar
  24. 24.
    Rosenberg, J.: Algebraic K-Theory and its Applications. Graduate Texts in Mathematics 147, Springer-Verlag, New York (1994)Google Scholar
  25. 25.
    Stenström, B.: Rings of Quotients. Die Grundlehren der Mathematischen Wissenschaften 217, Springer-Verlag, New York (1975)Google Scholar
  26. 26.
    Vaš , L.: Dimension and torsion theories for a class of Baer *-rings. J. Algebra 289(2), 614–639 (2005)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Vaš, L.: Class of Baer *-rings Defined by a Relaxed Set of Axioms. J. Algebra 297(2), 470–473 (2006)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Vaš, L.: A simplification of Morita’s construction of total right rings of quotients for a class of rings. J. Algebra 304(2), 989–1003 (2006)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Vaš, L.: Semisimplicity and global dimension of a finite von Neumann algebra. Math. Bohem. 132(1), 13–26 (2007)MathSciNetMATHGoogle Scholar
  30. 30.
    Vaš, L.: Perfect symmetric rings of quotients. J. Alg. Appl. 8(5), 689–711 (2009)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Physics and StatisticsUniversity of the Sciences in PhiladelphiaPhiladelphiaUSA

Personalised recommendations