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The ideal structure of existentially closed algebras

Published online by Cambridge University Press:  12 March 2014

Paul C. Eklof  and
Hans-Christian Mez
Paul C. Eklof
Affiliation:
Department of Mathematics, University of Californiaat Irvine Irvine, California 92717
Hans-Christian Mez
Affiliation:
Department of Mathematics, University of Californiaat Irvine Irvine, California 92717
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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).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 4 , December 1985 , pp. 1025 - 1043
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

[BT]Belyaev, V. Ya. and Taĭtslin, M. A., On elementary properties of existentially closed systems, Uspekhi Matematicheskikh Nauk, vol. 34 (1979), no. 2, pp. 39–94; English translation in Russian Mathematical Surveys, vol. 34 (1979), no. 2, pp. 43–107.Google Scholar
[B]Bokut′, L. A., Some embedding theorems for rings and semigroups, Sibirskiǐ Matematicheskiĭ Zhurnal, vol. 4 (1963), pp. 500–518, 729–743. (Russian) MR 27 # #2530, 3672.Google Scholar
[C1]Cohn, P. M., Universal algebra, 2nd ed., Reidel, Dordrecht, 1981.CrossRefGoogle Scholar
[C2]Cohn, P. M., On the free product of associative rings, Mathematische Zeitschrift, vol. 71 (1969), pp. 380–398.CrossRefGoogle Scholar
[E]Eklof, P. C., Ultraproducts for algebraists, Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 105–137.CrossRefGoogle Scholar
[EM]Eklof, P. C. and Mez, H.-C., Additive groups of existentially closed rings, to appear in the proceedings of the Udine conference on abelian groups and modules, CISM series.Google Scholar
[ES]Eklof, P. C. and Sabbagh, G., Model-completions and modules, Annals of Mathematical Logic, vol. 2 (1971), pp. 251–295.CrossRefGoogle Scholar
[F]Fuchs, L., Infinite abelian groups, vols. 1, 2, Academic Press, New York, 1970, 1973.Google Scholar
[Ga]Garavaglia, S., Dimension and rank in the model theory of modules, preprint.Google Scholar
[G]Goodearl, K. R., Von Neumann regular rings, Pitman, London, 1979.Google Scholar
[HW]Hirschfeld, J. and Wheeler, H. W., Forcing, arithmetic, and division rings, Lecture Notes in Mathematics, vol. 454, Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
[K]Kaplansky, I., Infinite abelian groups, rev. ed., University of Michigan Press, Ann Arbor, Michigan, 1969.Google Scholar
[Mc]McCoy, N. H., The theory of rings, reprint, Chelsea, New York, 1973.Google Scholar
[M1]Mez, H.-Ch., Existentiell abgeschlossene Ringe und Algebren, thesis, University of Freiburg, Freiburg, 1982.Google Scholar
[M2]Mez, H.-Ch., Combinatorial extensions of radical, nil and locally nilpotent algebras, Journal of Algebra (to appear).Google Scholar
[R]Rotman, J., An introduction to homological algebra, Academic Press, New York, 1979.Google Scholar
[RY]Rotman, J. and Yen, T., Modules over a complete discrete valuation ring, Transactions of the American Mathematical Society, vol. 98 (1961), pp. 242–254.CrossRefGoogle Scholar
[Sz]Szász, F. A., Radicals of rings, Wiley, New York, 1981.Google Scholar
[W1]Warfield, R. B. Jr., Purify and algebraic compactness for modules, Pacific Journal of Mathematics, vol. 28 (1969), pp. 699–719.CrossRefGoogle Scholar
[W2]Warfield, R. B. Jr., Decomposability of finitely presented modules, Proceedings of the American Mathematical Society, vol. 25 (1970), pp. 167–172.CrossRefGoogle Scholar
[Z]Ziegler, M., Model theory of modules, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 149–213.CrossRefGoogle Scholar