Generalized skew polynomial rings

Dauns, John

doi:10.1090/S0002-9947-1982-0654851-2

Generalized skew polynomial rings
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Trans. Amer. Math. Soc. 271 (1982), 575-586 Request permission

Abstract:

For a totally ordered cancellative semigroup $\Gamma$, a skew field $K$, let $K[\Gamma ;\phi ]$ be a skew semigroup ring. If $x \in \Gamma , k \in K$, then $kx = x{k^x}$, where $k \to {k^x}$ is an endomorphism of $K$ depending on $x$. Ideals of $K[\Gamma ;\phi ]$ are investigated for various semigroups or groups $\Gamma$.

References

P. M. Cohn, Free rings and their relations, London Mathematical Society Monographs, No. 2, Academic Press, London-New York, 1971. MR 0371938
Paul Moritz Cohn, Skew field constructions, London Mathematical Society Lecture Note Series, No. 27, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0463237
Paul Conrad, Ordered semigroups, Nagoya Math. J. 16 (1960), 51–64. MR 109850
Paul Conrad and John Dauns, An embedding theorem for lattice-ordered fields, Pacific J. Math. 30 (1969), 385–398. MR 246859
Paul Conrad and Paul McCarthy, The structure of $f$-algebras, Math. Nachr. 58 (1973), 169–191. MR 330000, DOI 10.1002/mana.19730580111
John Dauns, Integral domains that are not embeddable in division rings, Pacific J. Math. 34 (1970), 27–31. MR 279018
John Dauns, Power series semigroup rings, Pacific J. Math. 34 (1970), 365–369. MR 269569
John Dauns, Embeddings in division rings, Trans. Amer. Math. Soc. 150 (1970), 287–299. MR 262291, DOI 10.1090/S0002-9947-1970-0262291-5
John Dauns, Ordered domains, Symposia Mathematica, Vol. XXI (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), Academic Press, London, 1977, pp. 565–587. MR 0485619

Noncyclic division rings

169

John Dauns, Generalized semigroup rings, Algebra, Carbondale 1980 (Proc. Conf., Southern Illinois Univ., Carbondale, Ill., 1980) Lecture Notes in Math., vol. 848, Springer, Berlin, 1981, pp. 235–254. MR 613190
L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864

Teilweise geordnete algebraische Strukturen

Charles Holland, A totally ordered integral domain with a convex left ideal which is not an ideal, Proc. Amer. Math. Soc. 11 (1960), 703. MR 125140, DOI 10.1090/S0002-9939-1960-0125140-0
Nathan Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys, Vol. II, American Mathematical Society, New York, 1943. MR 0008601
Arun Vinayak Jategaonkar, Left principal ideal domains, J. Algebra 8 (1968), 148–155. MR 218387, DOI 10.1016/0021-8693(68)90040-9
A. I. Mal′cev, On the embedding of group algebras in division algebras, Doklady Akad. Nauk SSSR (N.S.) 60 (1948), 1499–1501 (Russian). MR 0025457
Susan Montgomery and D. S. Passman, Crossed products over prime rings, Israel J. Math. 31 (1978), no. 3-4, 224–256. MR 516150, DOI 10.1007/BF02761494
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
Wayne B. Powell, Projectives in a class of lattice ordered modules, Algebra Universalis 13 (1981), no. 1, 24–40. MR 631407, DOI 10.1007/BF02483820
M. Satyanarayana, Principal right ideal semigroups, J. London Math. Soc. (2) 3 (1971), 549–553. MR 283115, DOI 10.1112/jlms/s2-3.3.549
Stuart A. Steinberg, An embedding theorem for commutative lattice-ordered domains, Proc. Amer. Math. Soc. 31 (1972), 409–416. MR 285464, DOI 10.1090/S0002-9939-1972-0285464-1