$H_v$-rings first were introduced by Vougiouklis in 1990. The largest class of algebraic systems satisfying ring-like axioms is the $H_v$-ring. Let R be an $H_v$-ring and ${gamma}_R$ the smallest equivalence relation on R such that the quotient $R/{gamma}_R$, the set of all equivalence classes, is a ring. In this case $R/{gamma}_R$ is called the fundamental ring. In this short communication, we study the fundamental rings with respect to the product of two fuzzy subsets.