The ideal structure of existentially closed algebras

Throughout this paper, ⊿ will denote a commutative ring with multiplicative identity, 1. The algebras we consider will be associative ⊿-algebras which are not necessarily commutative and do not necessarily contain a multiplicative identity. By standard methods, every ⊿-algebra can be embedded in an existentially closed (e.c.) Δ-algebra—and even in one which is existentially universal (e.u.). (See §0 for more details.)We shall be studying the ideals of e.c. ⊿-algebras. Since every ideal is a sum of principal ideals, a natural place to begin is with principal ideals. In §1 we show that for an algebraically closed (a.c.) ⊿-algebra A, and elements a, b in A, whether or not b belongs to the principal ideal (a)A generated by a, depends only on the underlying ⊿-module structure of A; more precisely, for b to belong to (a)A it is necessary and sufficient that b satisfies every positive existential formula θ(ν) in the language of ⊿-modules which is satisfied by a (cf. Corollary 1.8). For special classes of rings ⊿ this condition can be simplified (Proposition 1.10): e.g. for Prüfer rings it is enough to consider formulas of the form ∃x(λx = μν); and for regular rings it is enough to consider formulas μν = 0 (where λ, μ ∈ ⊿).In §2 we use the results of §1 to study e.c. and e.u. algebras over a principal ideal domain (p.i.d.) ⊿ (Note that for ⊿ = Z this includes the case of e.c. rings.) We obtain a necessary and sufficient condition for an a.c. ⊿-algebra to be e.c. (Theorem 2.4). We also show (Theorem 2.2) that in an a.c. ⊿-algebra A every element that is divisible by all nonzero elements of ⊿ belongs to the divisible part D(A) of A. (It should be noted that, while a.c. ⊿-modules are always divisible [ES], an e.c. ⊿-algebra is never divisible: see the end of §0. Moreover, an e.c. ⊿-algebra always contains torsion-free elements: see Remark 2.3.) We prove that every bounded ideal in an a.c. ⊿-algebra is principal (2.7).