α-fuzzy Quotient Modules

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1 International Mathematical Forum, 4, 2009, no. 32, α-fuzzy Quotient Modules S. K. Bhambri and Pratibha Kumar Department of Mathematics Kirori Mal College (University of Delhi) Delhi , India Abstract. α-fuzzy cosets of a fuzzy submodule and α-fuzzy quotient modules are defined and discussed. A homomorphism from a given module onto the set of all α-fuzzy quotient module is established. The dimension of α-fuzzy quotient module is determined. Mathematics Subject Classification: 16Y99, 20N20 Keywords: Fuzzy set, fuzzy submodule, level subset, α-fuzzy coset, α-fuzzy quotient module 1. Introduction Throughout the paper, it is assumed that M is an R-module, where R is a commutative ring with unity, θ, a fuzzy submodule of M and α [0,θ(0)]. The concept of fuzzy set was first introduced by Zadeh [8] and then applied to the theory of groups by Rosenfeld [7] to that of semigroups by Kuroki [5]. The study of fuzzy submodules was later introduced by Pan [6] and Golan [1]. In this paper α-fuzzy coset of a fuzzy submodule is defined and the set S of all α-fuzzy cosets is shown to form a module under well defined operations on S. A fuzzy subset of S is defined and shown to form a fuzzy submodule, called α-fuzzy quotient module. It is further shown that S is a homomorphic image of an R-module M. Finally, the dimension of α-fuzzy quotient module is determined.

2 1556 S. K. Bhambri and P. Kumar 2. Preliminaries and Results 2.1. Definition. If μ is any fuzzy subset of a set T and t [0, 1], then the set μ t = {x T μ(x) t} is called a level subset of μ Definition ([4]). A fuzzy subset θ of M is called a fuzzy submodule of M if the following conditions are satisfied: (i) θ(x + y) min{θ(x),θ(y)}, for all x, y M; and (ii) θ(rx) θ(x), for all r R, x M Proposition. Let θ be a fuzzy subset of M. Then θ t = {x M θ(x) t}, t Imθ is a submodule of M if and only if θ is a fuzzy submodule of M Definition. Let θ be a fuzzy submodule of an R-module M. Let α [0,θ(0)]. For any x M define a fuzzy subset θx α of M called α-fuzzy coset of θ in M, as follows: θx α (m) = min{θ(x + m),α}, for all m M Proposition. Let S be the set of all α-fuzzy cosets of θ in M i.e., S = {θx α x M}. Then the two operations and defined on the set S as follows are well defined. (i) θx α θα y = θα x+y, for all x, y M. (ii) r θx α = θα rx, for all x M, r R. Proof. Let θx α = θα x and θα y = θα y ; x, x,y,y M. Let m M, then (θx α θα y )(m) = θα x+y (m) = min{θ(x + y + m),α} = θx α (y + m) = θx α (y + m) = min{θ(x + y + m),α} = min{θ(y +(x + m)),α} = θy α (x + m) = θy α (x + m) = min{θ(y + x + m),α} = θx α +y (m) = (θx α θα y )(m).

3 α-fuzzy quotient modules 1557 Therefore is well defined. Now, let θ α x = θ α x for x, x M. Then θ α x ( x )=θ α x ( x ). So that min(θ(x x ),α) = min(θ(x x ),α)=α. Therefore θ(x x ) α, so that x x θ α = N, say, that is θ α + x = θ α + x. Now we show that if θ α + x = θ α + x then θ α x = θα x. θ α + x = θ α + x implies θ α + x + y = θ α + x + y, for all y M. Suppose θ(x + y) <αand θ(x + y) α. θ(x + y) α implies x + y θ α that is θ α + x + y = θ α that is θ α + x + y = θ α so that x + y θ α and so θ(x + y) α, contradicting our assumption. Similarly θ(x + y) α and θ(x + y) <α, also leads to a contradiction. Therefore, either θ(x + y) α and θ(x + y) α or θ(x + y) <α and θ(x + y) <α. In the first case θx α (y) = min{θ(x + y),α} = α, and θ α x (y) = min{θ(x + y),α} = α. In the second case θ α x (y) = min{θ(x + y),α} = θ(x + y) <α, and θ α x (y) = min{θ(x + y),α} = θ(x + y) <α. Now since θ α + x = θ α + x, therefore x = n + x, where n θ α (i.e. θ(n) α).

4 1558 S. K. Bhambri and P. Kumar Thus θx α (y) = min{θ(x + y),α} = θ(x + y) = θ(x n + y) min{θ(n),θ(x + y)} = min{θ(x + y),α} = θx α (y) = min{θ(x + y),α} = θ(x + y) = θ(n + x + y) min{θ(x + y),θ(n)} = min{θ(x + y),α} = θx α (y). So that in any case θ α x (y) =θ α x (y). Consequently, θx α = θα x iff θ α + x = θ α + x iff θ α + rx = θ α + rx, r R iff θrx α = θα rx,iffr θα x = r θα x. Hence is well defined. The following result can be easily proved Proposition. The set S of all α-fuzzy cosets of a fuzzy submodule θ of M, forms a module under the well defined binary operations and Theorem. The fuzzy subset θ of S defined as θ (θa α ) = sup {θ(x) x M}, θa α=θα x is a fuzzy submodule of S, called α-fuzzy quotient module. Proof. Let a, b M and let β = θ (θa α) and γ = θ (θb α). Let ε>0 be given. Then there exists x M such that β ε<θ(x), θ α + x = θ α + a and y M such that γ ε<θ(y), θ α + y = θ α + b. Therefore θ α + x + y = θ α + a + b. So that θ(x + y) θ (θa α θα b ) (as in Proposition 2.5).

5 α-fuzzy quotient modules 1559 Now, θ(x + y) min(θ(x),θ(y)) = θ(x) (say) > β ε. Therefore β ε<θ(x + y) θ (θa α θα b ), for all ε>0. So that β θ (θa α θb α). Now two cases arise : (i) β γ, then β ε γ ε that is γ ε β ε<θ (θa α θα b ), for all ε>0 so that γ θ (θa α θb α). Therefore min(θ (θa α ),θ (θb α )) = min(β,γ) = γ θ (θa α θb α ), (ii) β<γ, then min(θ (θa α ),θ (θb α )) = min(β,γ)=β θ (θa α θα b ). Thus in any case θ (θa α θα b ) min(θ (θa α ),θ (θb α )). Now let β = θ (θa α) α + y = θ α + a}. Let ε>0 be given, then there exists y M such that β ε<θ(y), (where θ α + y = θ α + a, that is θ α + ry = θ α + ra). But θ (r θa α)=θ (θra α ) = sup{θ(x) θ α + x = θ α + ra}. Therefore β ε<θ(y) θ(ry) θ (θra α ). Hence β ε<θ (θra) α for all ε>0, whence β θ (θra). α Consequently, θ (r θa α )=θ (θra α ) θ (θa α ). (1) and (2) imply that θ is a fuzzy submodule of S Proposition. A mapping f : M S, where M is an R-module and S is the set of α-fuzzy cosets of the fuzzy submodule θ of M, defined as f(x) =θx α is an onto homomorphism with Kerf = θ α where α [0,θ(0)]. Proof. Clearly f is an onto homomorphism. Let x Kerf, then f(x) = zero of S = θ0 α. Therefore θx α = θα 0 so that θα x (0) = θα 0 (0) that is min(θ(x),α) = min(θ(0),α) = α (since α [0,θ(0)]).

6 1560 S. K. Bhambri and P. Kumar Thus θ(x) α, so that x θ α hence Kerf θ α. Conversely, let x θ α which implies θ α + x = θ α so that θ α + x + m = θ α + m for all m M. Suppose θ(x + m) <αand θ(m) α. θ(m) α implies m θ α so that θ α + m = θ α that is θ α + x + m = θ α. Therefore x + m θ α so that θ(x + m) α, a contradiction. Similarly θ(x + m) α, θ(m) <αis not possible. Therefore, either θ(x + m) α, θ(m) α or θ(x + m) <α, θ(m) <α. In the first case min(θ(x + m),α)=α = min(θ(m),α) which implies θx α(m) =θα 0 (m). In the second case min(θ(x + m),α)=θ(x + m) <α and min(θ(m),α) =θ(m) <α. Now θx α (m) = min(θ(x + m),α) = θ(x + m) min(θ(x),θ(m)) = θ(m), (since x θ α ) = min(θ(m),α) = θ0 α (m). Also θ0 α (m) = min{θ(m),α} = θ(m) = θ(x + m x) min(θ(x + m),θ(x)) = θ(x + m) (as x θ α ) = min(θ(x + m),α) = θx(m) α. Therefore θx α (m) =θ0 α (m) which implies that θx α = θ0 α that is f(x) = zero of S, so that x Kerf.

7 α-fuzzy quotient modules 1561 Hence θ α Kerf. Soθ α =Kerf Theorem. If f : M S is an onto homomorphism then f(θ) =θ. Proof. Let a M. Then (f(θ),f(a)) = sup x f 1 (f(a)) {θ(x)} = sup {θ(x)} θx α=θα a = θ (θa α )=θ (f(a)). Therefore f(θ) =θ Theorem. Let θ be a fuzzy submodule of M and θ, a fuzzy submodule of S, then θ α = {θα 0 }. Proof. θ (θ0 α ) = sup{(θ(x) θ α + x = θ α } θ(n), for all n θ α α. Therefore θ0 α θ α. Let θa α θ α, then θ (θa α ) = β α = sup{θ(x) θ α + x = θ α + a}. Let ε>0 be given, then there exists x M such that θ α + x = θ α + a, so that x a = n θ α and θ(x) >β ε α ε. Therefore θ(a) = θ(a + n n), n θ α { α if θ(a + n) α min{θ(a + n),θ(n)} = = θ(a + n) if θ(a + n) <α. Thus in any case θ(a) >α ε, for ε>0. Therefore θ(a) α. So that a θ α which implies θ α + a = θ α so that θa α = θ0 α. Hence θ α = {θα 0 } Theorem. Let dim M< and let θ be any fuzzy submodule of M. If f : M N is a homomorphism, then dim θ = dim(θ/k f ) + dim(f(θ)), where M and N are modules over a field F.

8 1562 S. K. Bhambri and P. Kumar In view of the above result, the following theorem follows immediately Theorem. Let dim M< and θ be any fuzzy submodule of M and let θ be the α-fuzzy quotient module determined by θ, then dim θ = dim(θ/θ α ) + dim θ. That is dim θ = dim θ dim(θ/θ α ). References [1] Golan, J. S., Making modules fuzzy, Fuzzy Sets and Systems, 32 (1989) [2] Kumar, I. J., P. K. Saxena and Pratibha Yadav, Fuzzy normal subgroups and fuzzy quotients, Fuzzy Sets and Systems, 46 (1992) [3] Kumar, R., On the dimension of a fuzzy subspace, Fuzzy Sets and Systems, 54 (1993) [4] Kumar, R., S. K. Bhambri, Pratibha Kumar, Fuzzy submodules : some analogues and deviations, Fuzzy Sets and Systems, 70 (1995) [5] Kuroki, N. K., On fuzzy semigroups, Information Sciences, 53 (1991) [6] Pan, Fu-Zheng, Fuzzy finitely generated modules, Fuzzy Sets and Systems, 21 (1987) [7] Rosenfeld, A., Fuzzy groups, J. Math. Anal. Appl. 35 (1971) [8] Zadeh, L. A., Fuzzy sets, Inform. and Control, 8 (1965) Received: November, 2008

- Fuzzy Subgroups. P.K. Sharma. Department of Mathematics, D.A.V. College, Jalandhar City, Punjab, India

- Fuzzy Subgroups. P.K. Sharma. Department of Mathematics, D.A.V. College, Jalandhar City, Punjab, India International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 47-59 Research India Publications http://www.ripublication.com - Fuzzy Subgroups P.K. Sharma Department