Abstract
A reduced Rickart ring is considered as a reduced ring R with an additional operation which associates to every element \(a \in R\) the single idempotent e such that the ideal eR is the right annihilator of a. We discuss some elementary properties of this operation, prove that a ring is reduced and Rickart if and only if it is isomorphic to an associate ring in the sense of I. Sussman (a certain subdirect product of domains with “enough” idempotents), and present several equational axiom systems for reduced Rickart rings.
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09 December 2019
Corrected are some misprints and Theorem 6.5 in the joint paper with I.��Cremer, ���Notes on reduced Rickart rings, I. Representation and equational axiomatizations���.
References
Abian, A.: Rings without nilpotent elements. Math. Čas. 25, 289–291 (1975)
Andrunakievich, V.A., Ryabuchin, YuM: Rings without nilpotent elements, and completely prime ideals. Dokl. Akad. Nauk SSSR 180, 9–11 (1968)
Chacron, M.: Direct product of division rings and a paper of Abian. Proc. Am. Math. Soc. 29, 259–262 (1971)
Chase, U.: A generalization of the ring of triangular matrices. Nagoya Math. J. 18, 13–25 (1961)
Cīrulis, J.: Relatively orthocomplemented skew nearlattices in Rickart rings. Demonstr. Math. 48, 492–507 (2015)
Cīrulis, J.: Extending the star order to Rickart rings. Linear Multilinear Algebra 64, 1498–1508 (2016)
Cīrulis, J.: The diamond partial order for strong Rickart rings. Linear Multilinear Algebra 65, 192–203 (2017)
Cornish, W.H.: The variety of commutative Rickart rings. Nanta Math. 5, 43–51 (1972)
Cornish, W.H.: Amalgamating commutative regular rings. Comment. Math. Univ. Carol. 18, 423–436 (1977)
Cremer, I.: On reduced Rickart rings (Latvian), Master thesis, Dept. Math., University of Latvia, p. 59. https://dspace.lu.lv/dspace/handle/7/33470 (2016). Accessed 2 Nov 2017
Cvetko-Vah, K., Leech, J.: Rings whose idempotents form a multiplicative set. Commun. Algebra 40, 3288–3307 (2012)
Endo, S.: Note on P.P. rings (A supplement to Hattori’s paper). Nagoya Math. J. 17, 167–170 (1960)
Faith, K.: Algebra II. Ring theor. Springer, Berlin (1976). (e.a.)
Foulis, D.J.: Baer *-semigroups. Proc. Am. Math. Soc. 11, 648–654 (1960)
Fraser, J.A., Nicholson, W.K.: Reduced p.p.-rings. Math. Japn 34, 715–725 (1989)
Guo, X.H., Shum, K.P.: On p.p.-rings which are reduced. Int. J. Math. Math. Sci. 4, 5 (2006). (art. ID g34694)
Hattori, A.: A foundation of torsion theory for modules over general rings. Nagoya Math. J. 17, 147–158 (1960)
Huh, Ch., Kim, H.K., Lee, Y.: p. p. rings and generalized p. p. rings. J. Pure Appl. Algebra 167, 37–52 (2002)
Janowitz, M.F.: A note on Rickart rings and semi-Boolean algebras. Algebra Univ. 6, 9–12 (1976)
Lam, T.Y.: Lectures on modules and rings. Springer, Berlin (1999)
Maeda, S.: On a ring whose principal right ideals are generated by idempotents of a lattice. J. Sci. Hirosh. Univ. Ser. A 24, 509–525 (1960)
McCoy, N.H.: Subdirect sums of rings. Bull. Am. Math. Soc. 53, 856–877 (1947)
Speed, T.P., Evans, M.W.: A note on commutative Baer rings. Austral. Math. 13, 1–6 (1972)
Sussman, I.: A generalization of Boolean rings. Math. Ann. 136, 326–338 (1958)
Sussman, I.: Ideal structure and semigroup domain decomposition of associate rings. Math. Ann. 140, 87–93 (1960)
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Cīrulis, J., Cremer, I. Notes on reduced Rickart rings, I. Beitr Algebra Geom 59, 375–389 (2018). https://doi.org/10.1007/s13366-017-0373-3
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DOI: https://doi.org/10.1007/s13366-017-0373-3