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## Archive of Issues

Iraq Al Diwaniyah
Year
2018
Volume
28
Issue
2
Pages
176-192
 Section Mathematics Title On $\mathcal{L}$-injective modules Author(-s) Mehdi A.R.a Affiliations University of Al-Qadisiyaha Abstract Let $\mathcal{M}=\{(M,N,f,Q)\mid M,N,Q\in R\text{-Mod}, \,N\leq M,\,f\in \text{Hom}_{R}(N,Q)\}$ and let $\mathcal{L}$ be a nonempty subclass of $\mathcal{M}.$ Jirásko introduced the concept of $\mathcal{L}$-injective module as a generalization of injective module as follows: a module $Q$ is said to be $\mathcal{L}$-injective if for each $(B,A,f,Q)\in \mathcal{L}$ there exists a homomorphism $g\colon B\rightarrow Q$ such that $g(a)=f(a),$ for all $a\in A$. The aim of this paper is to study $\mathcal{L}$-injective modules and some related concepts. Some characterizations of $\mathcal{L}$-injective modules are given. We present a version of Baer's criterion for $\mathcal{L}$-injectivity. The concepts of $\mathcal{L}$-$M$-injective module and $s$-$\mathcal{L}$-$M$-injective module are introduced as generalizations of $M$-injective modules and give some results about them. Our version of the generalized Fuchs criterion is given. We obtain conditions under which the class of $\mathcal{L}$-injective modules is closed under direct sums. Finally, we introduce and study the concept of $\sum$-$\mathcal{L}$-injectivity as a generalization of $\sum$-injectivity and $\sum$-$\tau$-injectivity. Keywords injective module, generalized Fuchs criterion, hereditary torsion theory, $t$-dense, preradical, natural class UDC 512.553.3 MSC 16D50, 16D10, 16S90 DOI 10.20537/vm180204 Received 3 February 2018 Language English Citation Mehdi A.R. On $\mathcal{L}$-injective modules, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 176-192. References Anderson F.W., Fuller K.R. Rings and categories of modules, New York: Springer, 1992. DOI: 10.1007/978-1-4612-4418-9 Azumaya G., Mbuntum F., Varadarajan K. On $M$-projective and $M$-injective modules, Pacific J. Math., 1975, vol. 59, no. 1, pp. 9-16. DOI: 10.2140/pjm.1975.59.9 Baer R. Abelian groups that are direct summands of every containing abelian group, Bull. Amer. Math. Soc., 1940, vol. 46, no. 10, pp. 800-806. DOI: 10.1090/S0002-9904-1940-07306-9 Beachy J.A. A generalization of injectivity, Pacific J. Math., 1972, vol. 41, no. 2, pp. 313-327. DOI: 10.2140/pjm.1972.41.313 Bland P.E. A note on quasi-divisible modules, Communications in Algebra, 1990, vol. 18, no. 6, pp. 1953-1959. DOI: 10.1080/00927879008824003 Charalambides S. Topics in Torsion Theory, Ph.D. Thesis, New Zealand: University of Otago, 2006. Crivei S. A note on $\tau$-quasi-injective modules, Stud. Univ. Babeş Bolyai Math., 2001, vol. 46, no. 3, pp. 33-39. http://www.cs.ubbcluj.ro/~studia-m/2001-3/CRIVEI.pdf Crivei S. Injective modules relative to torsion theories, Cluj-Napoca: EFES Publishing House, 2004. Dauns J. Classes of modules, Forum Mathematicum, 1991, vol. 3, no. 3, pp. 327-338. DOI: 10.1515/form.1991.3.327 Dauns J., Zhou Y. Classes of Modules, Boca Raton: Chapman and Hall/CRC, 2006. Faith C. Rings with ascending condition on annihilators, Nagoya Math. J., 1966, vol. 27, no. 1, pp. 179-191. DOI: 10.1017/S0027763000011983 Fuchs L. On quasi-injective modules, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1969, vol. 23, no. 4, pp. 541-546. Golan J.S. Torsion theories, New York: Longman Scientific and Technical, 1986. Jirásko J. Generalized injectivity, Comment. Math. Univ. Carolin., 1975, vol. 16, no. 4, pp. 621-636. Kasch F. Modules and rings, London: Academic Press Inc., 1982. Mohamed S.H., Müller B.J. Continuous and discrete modules, London: Cambridge University Press, 1990. DOI: 10.1017/CBO9780511600692 Page S.S., Zhou Y.Q. On direct sums of injective modules and chain conditions, Canad. J. Math., 1994, vol. 46, no. 3, pp. 634-647. DOI: 10.4153/CJM-1994-034-9 Smith P.F. Injective modules and their generalizations, Glasgow: University of Glasgow, 1997. Stenström B. Rings of quotients, Berlin: Springer-Verlag, 1975. DOI: 10.1007/978-3-642-66066-5 Yi O. On injective modules and locally nilpotent endomorphisms of injective modules, Math. J. Okayama Univ., 1998, vol. 40, no. 1, pp. 7-13. http://ousar.lib.okayama-u.ac.jp/33679 Full text