Abstract
A proof is given that not every linearly ordered associative (associative-commutative) ring is the o-image of a free associative (associative-commutative) ring for some ordering of the latter. There are also nilpotent linearly ordered rings which are not o-epimorphic images of free associative or free associative-commutative n-nilpotent rings for n≥ 4, no matter what ordering is used for the latter.
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L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, New York (1963).
E. Ya. Gabovich and O. A. Ivanova, o-Epimorphic Images of Ordered Free Semigroups of Certain Manifolds [in Russian], Ninth All-Union Algebra Colloquium, Summary of Reports, Gomel' (1968), pp. 52–53.
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Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 693–697, June, 1971.
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Ivanova, O.A. Linearly ordered rings which are not o-epimorphic images of ordered free rings. Mathematical Notes of the Academy of Sciences of the USSR 9, 402–404 (1971). https://doi.org/10.1007/BF01094584
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DOI: https://doi.org/10.1007/BF01094584