Abstract
Let R be a commutative (and associative) ring with unity and let L be a loop (roughly speaking, a loop is a group which is not necessarily associative, see Definition 3.1). The loop algebra of L over R was introduced in 1944 by R.H. Bruck (1944) as a means to obtain a family of examples of nonassociative algebras and is defined in a way similar to that of a group algebra; i.e., as the free R-module with basis L, with a multiplication induced distributively from the operation in L.
The author was partially supported by a research grant from CNPq., Proc. 300243/79-0(RN)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Akasaki, T. Idempotent Ideals in Integral Group Rings, J. Alg., 23, 343–346, 1972.
Akasaki, T. Idempotent Ideals in Integral Group Rings II, Arch. Math., 24, 126–128, 1973.
Artin, E. Geometric Algebra, Interscience, New York, 1957.
de Barros, L.G.X. Isomorphisms of rational loop algebras, Comm. Alg., 21(11), 3977–3993, 1993a.
de Barros, L.G.X. On Semisimple Alternative Loop Algebras, Comm. Alg., 21(11), 3995–4011, 1993b.
Brauer, R. Über Systeme Hypercomplexer Zahlen, Math. Z., 30, 79–107, 1929.
Brauer, R. and Noether, E. Über minimale Zerfällungskörper irreduzibler Darstellungen, Sitz. Preuss. Akad. Wiss. Berlin, 221–228, 1927.
Bruck, R.H. Some Results in the Theory of Linear Non-Associative Algebras, Trans. Amer. Math. Soc., 56, 141–199, 1944.
Chein, O. and Goodaire, E.G. Loops whose Loop Rings are Alternative, Comm. Alg., 14, 293–310, 1986.
Chein, O. and Goodaire, E.G. Loops whose Loop Rings in characteristic 2 are Alternative, Comm. Alg., 18(3), 659–688, 1990.
Goodaire, E.G. Alternative Loop Rings, Publ. Math. Debrecen, 30, 31–38, 1983.
Goodaire, E.G. Jespers, E. and Polcino Milies, C. Alternative Loop Rings, North Holland Math. Studies 184, Elsevier, Amsterdam, 1996.
Goodaire, E.G. and Polcino Milies, C. Isomorphisms of Integral Alternative Loop Rings, Rend. Circ. Mat. Palermo, XXXVII, 126–135, 1988.
Goodaire, E.G. and Polcino Milies, C. Torsion Units in Alternative Loop Rings, Proc. Amer. Math. Soc., 107, 7–15, 1989.
Goodaire, E.G. and Polcino Milies, C. Finite subloops of units in an alternative loop ring, Proc. Amer. Math. Soc., 124(4), 995–1002, 1996.
Goodaire, E.G. and Polcino Milies, C. On the Loop of Units of an Alternative Loop Ring, Nove J. Alg. Geom., 3(3), 199–208, 1995a.
Goodaire, E.G. and Polcino Milies, C. Ring Alternative Loops and their Loop Rings, Resenhas Inst. Mat. Est. Univ, São Paulo, 2(1), 47–82, 1995b.
Hall, M. Jr., The Theory of Groups, MacMillan, New York, 1959.
Jespers, E. Leal, G. and Polcino Milies, C. Classifying indecomposable RA loops, J. Alg., 176, 5057–5076, 1995.
Klinger, L. Construction of a Counterexample to a Conjecture of Zassenhaus, Comm. Alg., 19, 2303–2330, 1991.
Leal, G. and Polcino Milies, C. Isomorphic Group (and Loop) Algebras, J. Alg., 155, 195–210, 1993.
Merlini Giuliani, M.L. Loops de Moufang Lineares, PhD. Thesis, Instituto de Matematica e Estatística, Universidade de São Paulo, 1998.
Liebeck, M.W. The classification of finite simple Moufang loops, Math. Proc. Comb. phil. Soc., 102, 33–47, 1987.
Moufang, R. Alternativekörper und der Satz vom vollständigen Viersteit (Dg), Abh. Math. Sem. Univ. Hamburg 9, 207–222, 1933.
Noether, E. Nichtkommutative Algebra, Math. Z., 37, 513–541, 1933.
Paige, L.J. A theorem on commutative power associative loop algebras, Proc. Amer. Math. Soc., 6, 279–280, 1955.
Paige, L.J. A class of simple Moufang loops, Proc. Amer. Math. Soc., 7, 471–482, 1956.
Polcino Milies, C. The torsion product property in alternative algebras II, Comm. Alg., to appear.
Roggenkamp, K.W. Integral Group Rings of Solvable Finite Groups have no Idempotent Ideals, Arch. Math. 25, 125–128, 1974.
Sehgal, S.K. Topics in Group Rings, Marcel Dekker, New York, 1978.
Sehgal, S.K. Units in Integral Group Rings, Longman Scientific & Technical/Essex, 1993.
Smith, P.F. A Note on Idempotent Ideals in Group Rings, Arch. Math., XXVII, 22–27, 1976.
Weiss, A. Units in Integral Group Rings, J. Reine. Angew. Math., 415, 175–187, 1991.
Zatelli, A. Elementos Nilpotentes em Anéis de Loop Alternativos, PhD. Thesis, Instituto de Matematica e Estatística, Universidade de São Paulo, 1993.
Zhevlakov, K.A. Slin’ko, A.M. Shestakov, I.P. and Shirshov, A.I. Rings That Are Nearly Associative, Academic Press, New York, 1982.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Hindustan Book Agency (India) and Indian National Science Academy
About this chapter
Cite this chapter
Milies, C.P. (1999). Alternative Loop Rings and Related Topics. In: Passi, I.B.S. (eds) Algebra. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-80250-94-6_9
Download citation
DOI: https://doi.org/10.1007/978-93-80250-94-6_9
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-20-3
Online ISBN: 978-93-80250-94-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)