Year 2012,
Volume: 11 Issue: 11, 96 - 110, 01.06.2012
Abstract
Let R be a ring with identity. Given a positive integer n, a unitary
right R-module X is called n–injective provided, for every n-generated right
ideal A of R, every R-homomorphism φ : A → X can be lifted to R. In
this note we investigate this and related injectivity conditions and show that
there are many rings R which have an n–injective module which is not (n+1)–
injective.
Keywords
References
- F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York, 1974.
- J.-E. Bj¨ork, Rings satisfying certain chain conditions, J. Reine Angew. Math., (1970), 63-73.
- V. P. Camillo, A note on semi-hereditary rings, Arch. Math. (Basel), 24 (1973), 143.
- K. L. Fields, On the global dimension of residue rings, PaciŞc J. Math., 32 (1970), 345-349.
- S. Jİndrup, P.P. rings and finitely generated flat ideals, Proc. Amer. Math. Soc., 28(2) (1971), 431-435.
- L. S. Levy, Torsion-free and divisible modules over non-integral domains, Canad. J. Math., 15 (1963), 132-151.
- W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics 158, Cambridge Univ. Press, Cambridge, 2003.
- V. A. Puninskaya, On injectivity properties for modules over domains, in Fong Yuen, A. A. Mikhalev and E. Zelmanov, Eds., Lie Algebras, Rings and Related Topics, Springer (2000), pp. 164-170.
- A. Shamsuddin, n-injective and n-flat-modules, Comm. Algebra, 29(5) (2001), 2050.
- P. F. Smith, On injective and divisible modules, Arab. J. Sci. Eng., to appear. A. A. Tuganbaev, Semihereditary rings and FP-injective modules, J. Math. Sci. (New York), 112(6) (2002), 4736-4742.
- Xiaoxiang Zhang and Jianlong Chen, On n-semihereditary and n-coherent rings, Int. Electron. J. Algebra, 1 (2007), 1-10.
- Esperanza S´anchez Campos Departamento de ´Algebra Geometr´ıa y Topolog´ıa Universidad de M´alaga M´alaga, Spain e-mail: esperanz@agt.cie.uma.es Patrick F. Smith Department of Mathematics University of Glasgow Glasgow G12 8QW, Scotland UK e-mail: Patrick.Smith@glasgow.ac.uk
Year 2012,
Volume: 11 Issue: 11, 96 - 110, 01.06.2012
Abstract
References
- F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York, 1974.
- J.-E. Bj¨ork, Rings satisfying certain chain conditions, J. Reine Angew. Math., (1970), 63-73.
- V. P. Camillo, A note on semi-hereditary rings, Arch. Math. (Basel), 24 (1973), 143.
- K. L. Fields, On the global dimension of residue rings, PaciŞc J. Math., 32 (1970), 345-349.
- S. Jİndrup, P.P. rings and finitely generated flat ideals, Proc. Amer. Math. Soc., 28(2) (1971), 431-435.
- L. S. Levy, Torsion-free and divisible modules over non-integral domains, Canad. J. Math., 15 (1963), 132-151.
- W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics 158, Cambridge Univ. Press, Cambridge, 2003.
- V. A. Puninskaya, On injectivity properties for modules over domains, in Fong Yuen, A. A. Mikhalev and E. Zelmanov, Eds., Lie Algebras, Rings and Related Topics, Springer (2000), pp. 164-170.
- A. Shamsuddin, n-injective and n-flat-modules, Comm. Algebra, 29(5) (2001), 2050.
- P. F. Smith, On injective and divisible modules, Arab. J. Sci. Eng., to appear. A. A. Tuganbaev, Semihereditary rings and FP-injective modules, J. Math. Sci. (New York), 112(6) (2002), 4736-4742.
- Xiaoxiang Zhang and Jianlong Chen, On n-semihereditary and n-coherent rings, Int. Electron. J. Algebra, 1 (2007), 1-10.
- Esperanza S´anchez Campos Departamento de ´Algebra Geometr´ıa y Topolog´ıa Universidad de M´alaga M´alaga, Spain e-mail: esperanz@agt.cie.uma.es Patrick F. Smith Department of Mathematics University of Glasgow Glasgow G12 8QW, Scotland UK e-mail: Patrick.Smith@glasgow.ac.uk
There are 12 citations in total.
Details
| Other ID | JA96JV73VS |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | June 1, 2012 |
| Published in Issue | Year 2012 Volume: 11 Issue: 11 |