A note on fuzzy characteristic and fuzzy divisor of zero of a ring

A <strong>note</strong> on fuzzy characteristic . . . 433. Fuzzy Homomorphism and Fuzzy IsomorphismMordeson and Malik [3] introduced the concepts of fuzzy homomorphismsand fuzzy isomorphisms of rings as follows:Let θ ∈ [0, 1).Let R and S be nonempty sets. Consider the following conditions on a fuzzysubset f of the Cartesian product R × S:(1) ∀x ∈ R, ∃y ∈ S such that f(x, y) > θ;(2) ∀y ∈ S, ∃x ∈ R such that f(x, y) > θ;(3) ∀x ∈ R, ∀y 1 , y 2 ∈ S, f(x, y 1 ) > θ and f(x, y 2 ) > θ implies y 1 = y 2 ;(4) ∀x 1 , x 2 ∈ R, ∀y ∈ S, f(x 1 , y) > θ and f(x 2 , y) > θ implies x 1 = x 2 .Let f be a fuzzy subset of R × S. If conditions (1) and (3) hold, then f iscalled a fuzzy function of R into S. If conditions (1), (2) and (3) hold, then fis called a fuzzy function of R onto S. If conditions (1), (3) and (4) hold, thenf is called a one-to-one fuzzy function of R into S.A fuzzy function f from a ring R into a ring S is called fuzzy homomorphismif and only if∀x 1 , x 2 ∈ R , ∀y ∈ Sf(x 1 + x 2 , y) ≥ sup{min(f(x 1 , y 1 ), f(x 2 , y 2 ) : y 1 , y 2 ∈ S, y = y 1 + y 2 )};f(x 1 x 2 , y) ≥ sup{min(f(x 1 , y 1 ), f(x 2 , y 2 ) : y 1 , y 2 ∈ S, y = y 1 y 2 )}.If a fuzzy homomorphism from R onto S is one–to–one, then it is called afuzzy isomorphism of R onto S.Definition 3.1. Suppose f is a fuzzy homomorphism from a ring R onto a ringS and A is a fuzzy subset of R, then f is called a fuzzy isomorphism with respectto A iff∀x 1 , x 2 ∈ R , ∀y ∈ Sf(x 1 , y) > 0 and f(x 2 , y) > 0 implies A(x 1 ) = A(x 2 ) .Theorem 3.1. Suppose A is a fuzzy ideal of a ring R with identity e such thatA(e) > 0. If F C A (R) is 0, then the additive subgroup of R generated by theidentity e is fuzzy isomorphic to the integral domain Z of all integers.Proof. Let F C A (R) be 0. Then F O A (e) in the additive group of R is infinite.Let R ∗ de<strong>note</strong> the additive subgroup of R generated by e. Since A is a fuzzyideal of R we have 0 < A(e) ≤ A(x) ≤ A(o) for all x ∈ R. We also <strong>note</strong> that

44 A. K. RayA(ne) < A(o) for any n ∈ Z by assumption. We define a fuzzy subset f of theCartesian cross-product R ∗ × Z as follows:f(ne, n) = A(ne) for all integers n ∈ Z ,f(ne, m) = 0 for all integers n, m ∈ Z with n ≠ m .From the definition of f it clearly follows that f is a one–to–one fuzzy functionof R ∗ onto Z.Let n 1 e, n 2 e ∈ R ∗ and n ∈ Z. If n = n 1 + n 2 , thenf(n 1 e + n 2 e, n) = f((n 1 + n 2 )e, n 1 + n 2 ) = A(n 1 e + n 2 e)≥ inf(A(n 1 e), A(n 2 e)) = inf(f(n 1 e, n 1 ), f(n 2 e, n 2 )) .If n ≠ n 1 +n 2 , we suppose n = m 1 +m 2 , where m 1 , m 2 ∈ Z, then either n 1 ≠ m 1or n 2 ≠ m 2 . Let n 1 ≠ m 1 . In this case f(n 1 e + n 2 e), n) = 0 and f(n 1 e, m 1 ) = 0and so inf(f(n 1 e, m 1 ), f(n 2 e, m 2 )) = 0.Thus we find that ∀n 1 e, n 2 e ∈ R ∗ and ∀n ∈ Z,f((n 1 e)(n 2 e), n) ≥≥ sup{inf(f(n 1 e, m 1 ), f(n 2 e, m 2 )) : n = m 1 + m 2 , m 1 , m 2 ∈ Z} .Now if n = n 1 n 2 , thenf((n 1 e)(n 2 e), n) = f((n 1 n 2 )e, n 1 n 2 ) = A((n 1 n 2 )e) = A((n 1 e)(n 2 e))≥ sup(A(n 1 e), A(n 2 e)) = sup(f(n 1 e, n 1 ), f(n 2 e, n 2 )) .If n ≠ n 1 n 2 , suppose n = m 1 m 2 where m 1 , m 2 ∈ Z, then either n 1 ≠ m 1 orn 2 ≠ m 2 . In this case we observe f((n 1 e)(n 2 e), n) = 0 and f(n 1 e, m 1 ) = 0, ifn 1 ≠ m 1 . So inf(f(n 1 e, m 1 ), f(n 2 e, m 2 )) = 0. Thus we find that ∀n 1 e, n 2 e ∈ R ∗and ∀n ∈ Z,f((n 1 e)(n 2 e), n) ≥≥ sup{inf(f(n 1 e, m 1 ), f(n 2 e, m 2 )) : n = m 1 m 2 , m 1 , m 2 ∈ Z} .Hence f is a fuzzy isomorphism of R ∗ onto Z. Thus the proof is completed. ✷Theorem 3.2. Suppose A is a fuzzy ideal of a ring R with identity e such thatA(e) > 0 and A(e) ≠ A(o). If R contains no fuzzy divisor of zero with respectto A and F C A (R) ≠ 0, then R ∗ is fuzzy isomorphic with respect to A to thefield Zp of all integers mod p, for some prime p.Proof. Suppose R contains no fuzzy divisors of zero with respect to A andF C A (R) ≠ 0. Then F C A (R) ≠ 0 is a prime p. Since A(e) ≠ A(o), F O A (e) is p.We define a fuzzy subset g of the Cartesian cross-product R ∗ × Zp as follows:

A <strong>note</strong> on fuzzy characteristic . . . 45g(me,(n)) = A(ne), where n is the least no–negative integer obtained asremainder when m is divided by p and (n)∈ Zp, and g(me, (n 1 )) = 0 if n 1 ≠ n,(n 1 )∈ Zp.It can be easily verified that g is a fuzzy homomorphism from R ∗ onto Zp.If f(m 1 e,(n)) > 0 and f(m 2 e,(n)) > 0, then there exist r 1 , r 2 ∈ Z suchthat m 1 = r 1 p + n and m 2 = r 2 p + n. This implies that A(m 1 e − m 2 e) =A(((r 1 − r 2 )p)e) = A(o). Which implies that A(m 1 e) = A(m 2 e). Hence f is afuzzy isomorphism with respect to A of R ∗ onto Zp.✷Acknowledgement. The author is grateful to the referees for their valuable constructivesuggestions.References[1] Kim, J. G., Fuzzy orders relative to fuzzy subgroups. Information Sciences, 80(1994), 341–348.[2] Liu, W.–J., Fuzzy invariant subgroup and fuzzy ideals. Fuzzy Sets and Systems, 8(1982) 133–139.[3] Malik, D. S., Mordeson, J. N., Fuzzy homomorphisms of rings. Fuzzy Sets andSystems, 46 (1992), 139–146.[4] Novak, V., Fuzzy Sets and Their Applications. Bristol: Adam Hilger 1989.[5] Ray, A. K., Quotient group of a group generated by a subgroup and a fuzzy subset.Journal of Fuzzy Mathematics, 7(2) (1999), 459–463.Received by the editors September 2, 2002