Normal pairs of noncommutative rings

  • David E. Dobbs
  • Noômen JarbouiEmail author


This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if \(R \subseteq S\) are rings and \(D=(d_{ij})\) is an \(n\times n\) matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of \(n\times n\) matrices with entries in R if and only if each \(d_{ij}\) is integral over R. If \(R \subseteq S\) are rings with corresponding full rings of \(n\times n\) matrices \(R_n\) and \(S_n\), then \((R_n,S_n)\) is a normal pair if and only if (RS) is a normal pair. Examples are given of a pair \((\Lambda , \Gamma )\) of noncommutative (in fact, full matrix) rings such that \(\Lambda \subset \Gamma \) is (resp., is not) a minimal ring extension; it can be further arranged that \((\Lambda , \Gamma )\) is a normal pair or that \(\Lambda \subset \Gamma \) is an integral extension.


Associative ring Integrality Ring extension Normal pair Matrix Full matrix ring Minimal ring extension Prüfer domain Idealization 

Mathematics Subject Classification

Primary 16999 13B21 Secondary 13B99 13F05 13C15 



The authors wish to thank the referee for his/her very precise critique.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Al-Kuleab, N., Jarboui, N.: A note on intermediate matrix rings. Far East J. Math. Educ. 17(4), 227–229 (2018)CrossRefGoogle Scholar
  2. 2.
    Artin, M., Schelter, W.: Integral ring homomorphisms. Adv. Math. 39(3), 289–329 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atterton, T.W.: Definitions of integral elements and quotient rings over non-commutative rings with identity. J. Austral. Math. Soc. 13, 433–446 (1972)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ben Nasr, M., Jarboui, N.: Intermediate domains between a domain and some intersection of its localizations. Boll. Unione Mat. Ital. Sez. (8) 5–B(3), 701–713 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ben Nasr, M., Jarboui, N.: New results about normal pairs of rings with zero divisors. Ric. Mat. 63(1), 149–155 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourbaki, N.: Éléments de Mathématique, Algèbre Commutative, Chapitres 5–6, Actualités Scientifiques et Industrielles 1308, Hermann, Paris (1964)Google Scholar
  7. 7.
    Cohen, I.S., Seidenberg, A.: Prime ideals and integral dependence. Bull. Am. Math. Soc. 52, 252–261 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coykendall, J., Dobbs, D.E.: Survival-pairs of commutative rings have the lying-over property. Commun. Algebra 31(1), 259–270 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davis, E.D.: Overrings of commutative rings. II. Integrally closed overrings. Trans. Am. Math. Soc. 110, 196–212 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Davis, E.D.: Overrings of commutative rings. III. Normal pairs. Trans. Am. Math. Soc. 182, 175–185 (1973)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Demazure, M., Gabriel, P.: Introduction to Algebraic Geometry and Algebraic Groups, translated from the French by J. Bell, North-Holland Mathematical Studies 39. North-Holland, Amsterdam (1980)Google Scholar
  12. 12.
    Dobbs, D.E.: On INC-extensions and polynomials with unit content. Can. Math. Bull. 23(1), 37–42 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dobbs, D.E.: Lying-over pairs of commutative rings. Can. J. Math. 33, 454–475 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dobbs, D.E.: Every commutative ring has a minimal ring extension. Commun. Algebra 34, 3875–3881 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dobbs, D.E., Fontana, M., Papick, I.I.: On the flat spectral topology. Rend. Mat. 7(1), 559–578 (1981)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dobbs, D.E., Picavet, G., Picavet-L’Hermitte, M.: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebra 371, 391–429 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dobbs, D.E., Shapiro, J.: Normal pairs with zero-divisors. J. Algebra Appl. 10, 335–356 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dorroh, J.L.: Concerning adjunctions to algebras. Bull. Am. Math. Soc. 38, 85–88 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dorsey, T.J., Mesyan, Z.: On minimal extensions of rings. Commun. Algebra 37, 3463–3486 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ferrand, D., Olivier, J.-P.: Homomorphismes minimaux d’anneaux. J. Algebra 16, 461–471 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gilmer, R., Hoffmann, J.F.: A characterization of Prüfer domains in terms of polynomials. Pac. J. Math. 60(1), 81–85 (1975)CrossRefzbMATHGoogle Scholar
  22. 22.
    Herstein, I.N.: Noncommutative Rings. Carus Mathematical Monographs 15, Mathematical Association of America. Wiley, New York (1968)zbMATHGoogle Scholar
  23. 23.
    Huckaba, J.A.: Commutative Rings with Zero Divisors. Dekker, New York (1988)zbMATHGoogle Scholar
  24. 24.
    Jacobson, N.: Basic Algebra I. W. H. Freeman and Co., San Francisco (1974)zbMATHGoogle Scholar
  25. 25.
    Kaplansky, I.: Commutative Rings, rev edn. University of Chicago Press, Chicago (1974)zbMATHGoogle Scholar
  26. 26.
    Knebusch, M., Zhang, D.: Manis Valuations and Prüfer Extensions I. Lecture Notes on Mathematics 1791. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  27. 27.
    Paré, R., Schelter, W.: Finite extensions are integral. J. Algebra 53, 477–479 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Quinn, D.: Integrality over fixed rings. J. Lond. Math. Soc. 40(2), 206–214 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Quinn, D.: Integral extensions of noncommutative rings. Isr. J. Math. 73(1), 113–121 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Robson, J.C.: Some Results on Ring Extensions, edited by Christine Bessenrodt, Lecture Notes in Mathematics 4. Univ. Essen, Fachbereich Mathematik, Essen (1979)Google Scholar
  31. 31.
    Schelter, W.: Integral extensions of rings satisfying a polynomial identity. J. Algebra 40(1), 245–257 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  2. 2.Département de Mathématiques, Faculté des Sciences de SfaxUniversité de SfaxSfaxTunisia

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