On Prime A-ideals in Mv -modules

In 2003, Di Nola, et.al. introduced the notion of MV -modules over a PMV algebra and A-ideals in MV -modules [5]. These are structures that naturally correspond to lu-modules over lu-rings [5]. Recall that an lu-ring is a pair (R,u), where (R, ⊕, ·, 0, ≤) is an l-ring and u is a strong unit of R (i.e, u is a strong unit of the underlying l-group) such that u · u ≤ u and l-ring is a structure (R, +, ·, 0, ≤) that (R, +, 0, ≤) is an l-group such that for any x, y ∈ R, x ≥ 0 and y ≥ 0, we have x · y ≥ 0. They proved that the category of lu-modules over a given lu-ring (R, v) is equivalent to the category of MV -modules over Γ(R, v). They also proved there is a natural equivalence between MV -modules and truncated modules [5]. A. Dvurecenskij and A. Di Nola in [6] introduced the notion of PMV -algebras, that is MV -algebras whose product operation (·) is defined on the whole MV -algebra. This operation is associative and left/right distributive with respect to partially defined addition. They showed that the category of product MV -algebras is categorically equivalent to the category of associative unital l-rings. In addition, they introduced and studied MV F -algebras [6]. They also introduced ·-ideals in PMV -algebras. Then they showed that: Any MV F -algebra is a subdirect product of subdirectly irreducible MV F -algebras [6, Corollary 5.6]. Thus they concluded that a product MV -algebra is an MV F -ring if and only if it is a subdirect product of linearly ordered product MV -algebras [6, Theorem 5.8].