One of the foundational results of algebraic coding theory is the MacWilliams equivalence theorem. Wood has generalized this result to show that for finite Frobenius rings a generalization of the result holds. Determining what the class of rings to which the theorem will generalize remains an open question. The authors define a ring to be a MacWilliams ring if every linear isometry extends to a monomial transformation. A strategy is proposed which reduces the problem to questions about matrices over finite fields. Using these results they prove the known result that a commutative MacWilliams ring is Frobenius. Additionally they show that any left MacWilliams basic ring is Frobenius. The authors end by illustration some results with some concrete examples.