Semiring and Semimodule Issues in MV-Algebras

It is known that an atomic right LCM domain need not be a UFD but is a projectivity-UFD if it is also modular. This paper studies a slightly weaker and easier condition, the RAMP (acronym for the property in the title) , which also ensures that an atomic right LCM domain will be a projectivity-UFD. Among other things it is shown that in an atomic LCM domain, modularity is equivalent to the pair RAMP and LAMP (the left-right analog of RAMP). This result is then used to show that an atomic LCM domain with conjugation is modular. An example is given of an atomic LCM domain that has neither the RAMP nor the LAMP. All rings are not-necessarily commutative integral domains. Recall that an atomic ring is one in which every nonzero nonunit is a product of atoms (i.e. irreducibles) . A ring R is a right LCM domain if for any two elements a and b in R, aR ∩ bR is a principal right ideal. A right LCM domain need not be a left LCM domain 3] . If a ring has both properties it is called an LCM domain. It Is known (see Example 2 below) that, unlike the commutative case, an atomic right LCM domain need not be a UFD (unique factorization domain). In 1] it is shown that if the ring is also modular then it is a projectivity-UFD (definition of the latter recalled below)