MV-algebras derived from ideals in BL-algebras

We introduce the notion of ideal in BL-algebras as a natural generalization of that of ideal in MV-algebras. We show that from a purely algebraic point of view, the introduced notion has the proper meaning. Among other things, we establish that an ideal is prime if and only if the quotient BL-algebra is an MV-chain, and also that the MV-center of a BL-algebra as treated by Turunen and Sessa is simply the factor of the BL-algebra by the trivial ideal. We also analyze the relationship between ideals and deductive systems using the set of complement elements. It is our hope that this work will settle once and for all the existence of ideals in BL-algebras settings.