Osaka Journal of Mathematics

Almost relative injective modules

Surjeet Singh

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The concept of a module $M$ being almost $N$-injective, where $N$ is some module, was introduced by Baba (1989). For a given module $M$, the class of modules $N$, for which $M$ is almost $N$-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform, finite length module $U$ is almost $V$-injective, where $V$ is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of $V$. Let $M$ be a uniform module and $V$ be a finite direct sum of indecomposable modules. Some conditions under which $M$ is almost $V$-injective are determined, thereby Baba's result is generalized. A module $M$ that is almost $M$-injective is called an almost self-injective module. Commutative indecomposable rings and von Neumann regular rings that are almost self-injective are studied. It is proved that any minimal right ideal of a von Neumann regular, almost right self-injective ring, is injective. This result is used to give an example of a von Neumann regular ring that is not almost right self-injective.

Article information

Osaka J. Math., Volume 53, Number 2 (2016), 425-438.

First available in Project Euclid: 27 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16D50: Injective modules, self-injective rings [See also 16L60]
Secondary: 16E50: von Neumann regular rings and generalizations


Singh, Surjeet. Almost relative injective modules. Osaka J. Math. 53 (2016), no. 2, 425--438.

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  • A. Alahmadi and S.K. Jain: A note on almost injective modules, Math. J. Okayama Univ. 51 (2009), 101–109.
  • A. Alahmadi, S.K. Jain and S. Singh: Characterizations of almost injective modules; in Noncommutative Rings and Their Applications, Contemp. Math. 634, Amer. Math. Soc., Providence, RI, 11–17.
  • F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Springer, New York, 1974.
  • Y. Baba: Note on almost $M$-injectives, Osaka J. Math. 26 (1989), 687–698.
  • Y. Baba and M. Harada: On almost $M$-projectives and almost $M$-injectives, Tsukuba J. Math. 14 (1990), 53–69.
  • C. Faith: Algebra, II, Springer, Berlin, 1976.
  • K.R. Goodearl: Von Neumann Regular Rings, Monographs and Studies in Mathematics 4, Pitman, Boston, MA, 1979.
  • M. Harada: On modules with extending properties, Osaka J. Math. 19 (1982), 203–215.
  • M. Harada: On almost relative injectives on Artinian modules, Osaka J. Math. 27 (1990), 963–971.
  • \begingroup M. Harada: Direct sums of almost relative injective modules, Osaka J. Math. 28 (1991), 751–758. \endgroup
  • M. Harada: Note on almost relative projectives and almost relative injectives, Osaka J. Math. 29 (1992), 435–446.
  • M. Harada: Almost Relative Projective Modules and Almost Relative Injective Modules, monograph.