We define T-intrinsic product of fuzzy subsets of an algebraic object more general than a ring and study some of its fundamental properties. Deep connection that exists between T-intrinsic product and fuzzy subsets of a ring is brought to the surface. The concepts of T-fuzzy subring, T-fuzzy ideal and T-fuzzy subgroup are redefined, to include quite larger classes within their fold. Interconnection between these classes and T-intrinsic product is studied in detail. In the end we isolate different subclasses of the subsemigroup F(R) and reveal their ideal structure.
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