There are some interesting relations between submodules of a module and its intuitionistic fuzzy (IF) submodules. In this paper we investigate some relationshipsbetween submodules of a module and its IFsubmodules. Then we introduce a graph structure on IFsubmodules of a module and obtain some properties of it, that is the main goal of this paper. We define the intersection graph of submodules of a module M(G) and we show that a submodule Nof Mis a center in MGif and only if IFNis a center in IFG. We get some relationships between IFsubmodules of a module and their supports, as vertices of IFgraph and crisp graph of a module M, respectively. We show that an IFsubmodule Aof Mis center in IFgraph of Mif and only ifAis a center in crisp graph of M.In prime ring R, we show that every vertex of intersection graph of IFideals of Ris center. In general the nature of intersection graph of IFsubmodules of a module under intersection, homomorphic images, finite sum and other algebraic operations of its vertices, are investigated.
The concept of an intuitionistic fuzzy set, which is a generalisation of fuzzy set, was introduced by K. T. Atanassov in 1986. In this paper using of intuitionistic fuzzy small submodules we get some results about this kind of fuzzy submodules. First we give some preliminary properties of intuitionistic fuzzy submodules. Then we attempt to investigate various properties of intuitionistic fuzzy small submodules. A necessary and sufficient condition for intuitionistic fuzzy small submodules is established. We investigate the nature of intuitionistic fuzzy small submodules under direct sum. Also we study on relation between intuitionistic fuzzy small submodules and level subsets of them, and get some interesting results in this sense.
Let M be a module. Then M is called a D11 module if any submodule of M has a supplement which is a direct summand of M. Also M is called [Formula: see text] if every direct summand of M is D11. In this paper we investigate generalizations of D11 and [Formula: see text] modules, namely δ-D11 and [Formula: see text] modules. We will prove that any δ-D11 module M has a decomposition M = M1 ⊕ M2 with δ(M1) ≪δ M1 and δ(M2) = M2.
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