Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T06:39:31.978Z Has data issue: false hasContentIssue false
  • English
  • Français

RING ENDOMORPHISMS WITH LARGE IMAGES

Published online by Cambridge University Press:  25 February 2013

ANDRÉ LEROY  and
JERZY MATCZUK
ANDRÉ LEROY
Affiliation:
Université d'Artois, Faculté Jean Perrin Rue Jean Souvraz 62 307 Lens, France e-mail: leroy@euler.univ-artois.fr
JERZY MATCZUK
Affiliation:
Institute of Mathematics, Warsaw University Banacha 2, 02-097 Warsaw, Poland e-mail: jmatczuk@mimuw.edu.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism σ of a prime one-sided noetherian ring R is injective whenever the image σ(R) contains an essential left ideal L of R. If, in addition, σ(L)=L, then σ is an automorphism of R. Examples showing that the assumptions imposed on R cannot be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.

Keywords

16W2016P60
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 2 , May 2013 , pp. 381 - 390
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013

References

REFERENCES

1.Bass, H., Connell, E. H. and Wright, D., The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bull. Am. Math. Soc. 7 (2) (1982), 287–330.Google Scholar
2.Beidar, K. L., Fong, Y., Lee, P.-H. and Wong, T.-L., On additive maps of prime rings satisfying the Engel condition, Comm. Algebra 25 (12) (1997), 3889–3902.CrossRefGoogle Scholar
3.Bresar, M., One-sided ideals and derivations of prime rings, Proc. Am. Math. Soc. 122 (4) (1994), 979–983.Google Scholar
4.Bresar, M., Chebotar, M. A. and Martindale, W. S., Functional identities (Birkhäuser, Basel, Germany, 2007).CrossRefGoogle Scholar
5.Bresar, M., Martindale, W. S. 3rd and Miers, R. C., Maps preserving n-th powers, Comm. Algebra 26 (1) (1998), 117–138.Google Scholar
6.Hiremath, V. A., Hopfian rings and Hopfian modules, Indian J. Pure App. Math. 17 (7) (1986), 895–900.Google Scholar
7.Jategaonkar, A. V., Skew polynomial rings over orders in Artinian rings, J. Algebra 21 (1) (1972), 51–59.Google Scholar
8.Jordan, D. A., Bijective extensions of injective rings endomorphisms, J. Lond. Math. Soc. 25 (3) (1982), 435–448.CrossRefGoogle Scholar
9.Leroy, A. and Matczuk, J., Dérivations et automorphismes algébriques d'anneaux premiers, Comm. Algebra 13 (6) (1985), 1245–1266.Google Scholar
10.Matczuk, J., S-Cohn-Jordan extensions, Comm. Algebra 35 (3) (2007), 725–746.CrossRefGoogle Scholar
11.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings, Graduate Studies in Mathematics, vol. 30 (American Mathematical Society, Providence, RI, 2001).Google Scholar
12.Tripathi, S. P., On the hopficity of the polynomial rings, Proc. Indian Acad. Sci. 108 (2) (1998), 133–136.Google Scholar
13.Varadarajan, K., Hopfian and co-Hopfian objects, Publicacions Matemàtiques 36 (1992), 293–317.Google Scholar
14.Varadarajan, K., Study of Hopficity in certain classes of rings, Comm. Algebra 28 (2) (2000), 771–783.CrossRefGoogle Scholar
15.Varadarajan, K., Some recent results on hopficity co-hopficity and related properties, in International symposium on ring theory (Birkenmeier, G. F., Park, J. K., Park, Y. S., Editors) (Trends in Mathematics series) (Birkhäuser, Boston, MA, 2001), 371–392.Google Scholar