Notes on reduced Rickart rings, I

Representation and equational axiomatizations
  • Jānis CīrulisEmail author
  • Insa Cremer
Original Paper


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.


Equational axioms Focal operation Reduced ring Rickart ring Subdirect product Sussman ring 

Mathematics Subject Classification

16W99 16S60 16R10 



The authors are grateful to the anonymous referee for his/her remarks concerning the presentation and for suggestions that have resulted in Example 2.1 and Proposition 2.3. The first author was partially supported by Latvian Council of Science, Grant No. 271/2012.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest. The first author has not any financial relationship with the supporting organization.


  1. Abian, A.: Rings without nilpotent elements. Math. Čas. 25, 289–291 (1975)MathSciNetzbMATHGoogle Scholar
  2. Andrunakievich, V.A., Ryabuchin, YuM: Rings without nilpotent elements, and completely prime ideals. Dokl. Akad. Nauk SSSR 180, 9–11 (1968)MathSciNetGoogle Scholar
  3. Chacron, M.: Direct product of division rings and a paper of Abian. Proc. Am. Math. Soc. 29, 259–262 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Chase, U.: A generalization of the ring of triangular matrices. Nagoya Math. J. 18, 13–25 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cīrulis, J.: Relatively orthocomplemented skew nearlattices in Rickart rings. Demonstr. Math. 48, 492–507 (2015)MathSciNetzbMATHGoogle Scholar
  6. Cīrulis, J.: Extending the star order to Rickart rings. Linear Multilinear Algebra 64, 1498–1508 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cīrulis, J.: The diamond partial order for strong Rickart rings. Linear Multilinear Algebra 65, 192–203 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cornish, W.H.: The variety of commutative Rickart rings. Nanta Math. 5, 43–51 (1972)MathSciNetzbMATHGoogle Scholar
  9. Cornish, W.H.: Amalgamating commutative regular rings. Comment. Math. Univ. Carol. 18, 423–436 (1977)MathSciNetzbMATHGoogle Scholar
  10. Cremer, I.: On reduced Rickart rings (Latvian), Master thesis, Dept. Math., University of Latvia, p. 59. (2016). Accessed 2 Nov 2017
  11. Cvetko-Vah, K., Leech, J.: Rings whose idempotents form a multiplicative set. Commun. Algebra 40, 3288–3307 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Endo, S.: Note on P.P. rings (A supplement to Hattori’s paper). Nagoya Math. J. 17, 167–170 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Faith, K.: Algebra II. Ring theor. Springer, Berlin (1976). (e.a.)CrossRefzbMATHGoogle Scholar
  14. Foulis, D.J.: Baer *-semigroups. Proc. Am. Math. Soc. 11, 648–654 (1960)MathSciNetzbMATHGoogle Scholar
  15. Fraser, J.A., Nicholson, W.K.: Reduced p.p.-rings. Math. Japn 34, 715–725 (1989)MathSciNetzbMATHGoogle Scholar
  16. Guo, X.H., Shum, K.P.: On p.p.-rings which are reduced. Int. J. Math. Math. Sci. 4, 5 (2006). (art. ID g34694)MathSciNetzbMATHGoogle Scholar
  17. Hattori, A.: A foundation of torsion theory for modules over general rings. Nagoya Math. J. 17, 147–158 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Huh, Ch., Kim, H.K., Lee, Y.: p. p. rings and generalized p. p. rings. J. Pure Appl. Algebra 167, 37–52 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Janowitz, M.F.: A note on Rickart rings and semi-Boolean algebras. Algebra Univ. 6, 9–12 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Lam, T.Y.: Lectures on modules and rings. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  21. 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)zbMATHGoogle Scholar
  22. McCoy, N.H.: Subdirect sums of rings. Bull. Am. Math. Soc. 53, 856–877 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Speed, T.P., Evans, M.W.: A note on commutative Baer rings. Austral. Math. 13, 1–6 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Sussman, I.: A generalization of Boolean rings. Math. Ann. 136, 326–338 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Sussman, I.: Ideal structure and semigroup domain decomposition of associate rings. Math. Ann. 140, 87–93 (1960)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2017

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsUniversity of LatviaRigaLatvia
  2. 2.Department of MathematicsUniversity of LatviaRigaLatvia

Personalised recommendations