Strongly Semihereditary Rings and Rings with Dimension
 Lia Vaš
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The existence of a wellbehaved 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 threefold: (1) We study assumptions on a ring with involution that guarantee the existence of a wellbehaved 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 wellbehaved 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 Rmodule 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.
 Abrams, G, Aranda Pino, G (2005) The Leavitt path algebra of a graph. J. Algebra 293: pp. 319334 CrossRef
 Abrams, G, Aranda Pino, G, Siles Molina, M (2008) Locally finite Leavitt path algebras. Isr. J. Math. 165: pp. 329348 CrossRef
 Abrams, G, Tomforde, M (2011) Isomorphism and Morita equivalence of graph algebras. Trans. Am. Math. Soc. 363: pp. 37333767 CrossRef
 Ara, P, Brustenga, M (2007) The regular algebra of a quiver. J. Algebra 309: pp. 207235 CrossRef
 Ara, P, Menal, P (1984) On regular rings with involution. Arch. Math. Basel 42: pp. 126130 CrossRef
 Ara, P, Moreno, MA, Pardo, E (2007) Nonstable Ktheory for graph algebras. Algebr. Represent. Theory 10: pp. 157178 CrossRef
 Aranda Pino, G., Rangaswamy, K.L., Vaš, L.: *regular Leavitt path algebra of arbitrary graphs. Acta Math. Sin. (Eng. Ser.) (to appear, 2011)
 Aranda Pino, G., Vaš, L.: Noetherian Leavitt path algebras and their regular algebras. (preprint, 2011)
 Berberian, SK (1972) Baer *rings. Die Grundlehren der mathematischen Wissenschaften 195. SpringerVerlag, New York
 Berberian, S.K.: Baer rings and Baer *rings. (preprint, 1988) Available at www.ma.utexas.edu/mp_arc/c/03/03181.pdf
 Dung, NV, Huynh, DV, Smith, PF, Wisbauer, R (1994) Extending Modules. Pitman, London
 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)
 Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger, Malabar, FL (1991)
 Goodearl, KR (1971) Embedding nonsingular modules in free modules. J. Pure Appl. Algebra 1: pp. 275279 CrossRef
 Huynh, DV, Rizvi, ST, Yousif, MF (1996) Rings whose finitely generated modules are extending. J. Pure Appl. Algebra 111: pp. 325328 CrossRef
 Kaplansky, I (1968) Rings of Operators. Benjamin, New York
 Lam, TY (1999) Lectures on Modules and Rings. SpringerVerlag, New York CrossRef
 Lück, W.: L ^{2}invariants: Theory and Applications to Geometry and Ktheory. Ergebnisse der Mathematik und ihrer Grebzgebiete, Folge 3, 44, SpringerVerlag, Berlin (2002)
 Lück, W (1998) Dimension theory of arbitrary modules over finite von Neumann algebras and L 2Betti numbers I: Foundations. J. Reine Angew. Math. 495: pp. 135162
 Maeda, S, Holland, SS (1976) Equivalence of projections in Baer *rings. J. Algebra 39: pp. 150159 CrossRef
 Ortega, E (2008) Twosided localization of bimodules. Commun. Algebra 36: pp. 19111926 CrossRef
 Osofsky, BL, Smith, PF (1991) Cyclic modules whose quotients have all complement submodules direct summands. J. Algebra 139: pp. 342354 CrossRef
 Pyle, ES (1975) The regular ring and the maximal ring of quotients of a finite Baer *ring. Trans. Am. Math. Soc. 203: pp. 201213
 Rosenberg, J (1994) Algebraic KTheory and its Applications. Graduate Texts in Mathematics 147. SpringerVerlag, New York
 Stenström, B (1975) Rings of Quotients. Die Grundlehren der Mathematischen Wissenschaften 217. SpringerVerlag, New York
 Vaš, L (2005) Dimension and torsion theories for a class of Baer *rings. J. Algebra 289: pp. 614639 CrossRef
 Vaš, L (2006) Class of Baer *rings Defined by a Relaxed Set of Axioms. J. Algebra 297: pp. 470473 CrossRef
 Vaš, L (2006) A simplification of Morita’s construction of total right rings of quotients for a class of rings. Algebra 304: pp. 9891003 CrossRef
 Vaš, L (2007) Semisimplicity and global dimension of a finite von Neumann algebra. Math. Bohem. 132: pp. 1326
 Vaš, L (2009) Perfect symmetric rings of quotients. J. Alg. Appl. 8: pp. 689711 CrossRef
 Title
 Strongly Semihereditary Rings and Rings with Dimension
 Journal

Algebras and Representation Theory
Volume 15, Issue 6 , pp 10491079
 Cover Date
 20121201
 DOI
 10.1007/s1046801192791
 Print ISSN
 1386923X
 Online ISSN
 15729079
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Dimension
 Rings of quotients
 Semihereditary
 Involution
 Baer
 Regular
 16W99
 16S99 16S90 16W10
 Authors

 Lia Vaš ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Physics and Statistics, University of the Sciences in Philadelphia, Philadelphia, PA, 19104, USA