Fuzzy relations on rings and groups

Let S be any nonempty set. A fuzzy relation on S is a fuzzy subset of S × S. In this paper we study fuzzy relations on rings and groups. In particular we show that if μ and σ are fuzzy left (right) ideals of a ring R, then μ × σ is a fuzzy left (right) ideal of R × R and conversely if μ × σ is a fuzzy left (right) ideal of R × R, then either μ or σ is a fuzzy left (right) ideal of R. An example is given to show that if μ × σ is a fuzzy left (right) ideal of R × R, then μ and σ both need not be fuzzy left (right) ideals of R. An example is also given to show that if μ is a fuzzy left (right) ideal of R × R, then σ_μ, the weakest fuzzy subset of R on which μ is a fuzzy relation, need not be a fuzzy left (right) ideal of R. We obtain similar results for groups. We also show that certain results of Bhattacharya and Mukherjee (1985) are not true.