Fuzzy projective modules and tensor products in fuzzy module categories

Document Type : Research Paper


School of Mathematical Sciences, Shandong Normal University, 250014, Jinan, P. R. China


Let $R$ be a commutative ring. We write $\mbox{Hom}(\mu_A, \nu_B)$ for the set of all fuzzy $R$-morphisms from $\mu_A$ to $\nu_B$, where $\mu_A$ and $\nu_B$ are two fuzzy $R$-modules. We make
$\mbox{Hom}(\mu_A, \nu_B)$ into fuzzy $R$-module by redefining a function $\alpha:\mbox{Hom}(\mu_A, \nu_B)\longrightarrow [0,1]$. We study the properties of the functor $\mbox{Hom}(\mu_A,-):FR\mbox{-Mod}\rightarrow FR\mbox{-Mod}$ and get some unexpected results. In addition, we prove that
$\mbox{Hom}(\xi_p,-)$ is exact if and only if $\xi_P$ is a fuzzy projective $R$-module, when $R$ is a commutative semiperfect ring.
Finally, we investigate tensor product of two fuzzy $R$-modules and get some related properties. Also, we study the relationships between Hom functor and tensor functor.


\bibitem{Ameri} R. Ameri and M. M. Zahedi, {\it Fuzzy chain complex
and fuzzy homotopy}, Fuzzy sets and Systems, {\bf 112} (2000), 287-297.
\bibitem{AF} F. W. Anderson and K. R. Fuller, {\it Rings and Categories of Modules}, GTM13, Springer-Verlag, 1974.
\bibitem {Chen} Y. Chen, {\it Projective $S$-acts and exact functors}, Algebra Colloquium, {\bf 7(1)} (2000),
\bibitem {Gierz} G. Gierz, K. H. Hofmann, and etc., {\it Continuous Lattices and Domains},
Cambridge University Press, 2003.
\bibitem {Isaac}P. Isaac, {\it On projective L-modules}, Iranian Journal of Fuzzy
Systems, {\bf 2(1)} (2005), 19-28.
\bibitem{Lopez} S. R. L\'{o}pez-Permouth, {\it Lifting Morita equivalence to
categories of fuzzy modules}, Information Sciences, {\bf 64} (1992), 191-201.
\bibitem{LM} S. R. L\'{o}pez-Permouth and D. S. Malik, {\it On categories of fuzzy modules}, Information Sciences, {\bf 52} (1990), 211-220. 
\bibitem {Liu} H. Liu, {\it Hom functors and tensor product functors in fuzzy $S$-act category}, Neural Comput. \& Applic.,
{\bf 21(Suppl 1)} (2012), 275-279.
\bibitem {NR} C. V. Negoita and D. A. Ralescu,
{\it Applications of fuzzy subsets to system analysis}, Birkhauser,
Basel, 1975.
\bibitem {Pan-1} F. Pan, {\it Hom functors in the fuzzy category $F_m$},
Fuzzy Sets and Systems, {\bf 103} (1999), 525-528.
\bibitem {Pan-2} F. Pan, {\it The two functors in the fuzzy modular category}, Acta Mathematica
Scientia, {\bf 21B(4)} (2001), 526-530.
\bibitem {Rosen} A. Rosenfeld, {\it Fuzzy groups}, J. Math. Anal.
Appl., {\bf 35} (1971), 312-317.
\bibitem {Rotman} J. J. Rotman, {\it An introduction to homological algebra}, 2nd ed., Berlin: Springer,
\bibitem {Wyler} O. Wyler, {\it On the categories of general topology and topological
algebra}, Arch. der Math., {\bf 22} (1971), 7-17.
\bibitem {Zadeh} L. A. Zadeh, {\it Fuzzy sets}, Information and Control, {\bf 8} (1965),