A problem of annihilator primes in MVMV-algebras

Using Mundici’s equivalence [D. Mundici, Interpretation of AF C∗C∗-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1) (1986) 15–63] this note generalizes the result of [R. Cignoli, A. Torrens, The poset of prime ll-ideals of an abelian ll-group with strong unit, J. Algebra 184 (2) (1996) 604–612] for prime ideals and polars. Moreover, the generalization characterizes MVMV-algebras for which every prime ideal is the annihilator of a linear ordered ideal, and thus gives a negative answer to a problem of [L.P., Belluce, Semisimple algebras of infinite valued logic and bold fuzzy set theory, Canad. J. Math. 38 (6) (1986) 1356–1379]