Units in alternative integral loop rings. (English) Zbl 0881.20028
Given the ring \(\mathbb{Z}\) of integers and a loop \(L\) the integral loop ring \(\mathbb{Z} L\) is the free \(\mathbb{Z}\)-module with basis \(L\) and multiplication defined distributively from the multiplication of \(L\). In a ring a unit is an element which is invertible. \(RA\) loops are loops whose loop rings over any associative ring of characteristic \(\neq 2\) are alternative but not associative. A subloop \(H\) of an \(RA\) loop \(L\) is torsion if all elements in \(H\) are of finite order.
The authors study the “isomorphism problem”: When the ring isomorphism \(\mathbb{Z} L\cong\mathbb{Z} M\) implies the loop isomorphism \(L\cong M\)? They prove that the integral loop ring of a finitely generated \(RA\) loop determines the loop, i.e. \(\mathbb{Z} L\cong\mathbb{Z} M\Rightarrow L\cong M\). Also they deduce some properties of the torsion units of the integral loop ring \(\mathbb{Z} L\) of a finitely generated \(RA\) loop \(L\).
The authors study the “isomorphism problem”: When the ring isomorphism \(\mathbb{Z} L\cong\mathbb{Z} M\) implies the loop isomorphism \(L\cong M\)? They prove that the integral loop ring of a finitely generated \(RA\) loop determines the loop, i.e. \(\mathbb{Z} L\cong\mathbb{Z} M\Rightarrow L\cong M\). Also they deduce some properties of the torsion units of the integral loop ring \(\mathbb{Z} L\) of a finitely generated \(RA\) loop \(L\).
Reviewer: E.Brožíková (Praha)
MSC:
| 20N05 | Loops, quasigroups |
| 17D05 | Alternative rings |
| 16U60 | Units, groups of units (associative rings and algebras) |
Keywords:
isomorphism problem; integral loop rings; \(RA\) loops; ring isomorphisms; loop isomorphisms; torsion unitsReferences:
| [1] | L.G.X. de Barros and S.O. Juriaans, Units in Integral Loop Rings, J. Algebra 183 (1996), 637–648. · Zbl 0874.20045 · doi:10.1006/jabr.1996.0236 |
| [2] | A.A. Bovdi, Z.S. Marciniak, S.K. Sehgal, Torsion Units in Infinite Group Rings, J. Number Theory 47 (1994), 284–299. · Zbl 0804.16036 · doi:10.1006/jnth.1994.1038 |
| [3] | O. Chein and E.G. Goodaire, Loops whose Loop Rings are Alternative, Comm, in Algebra, 14 (1986), 293–310. · Zbl 0582.17015 · doi:10.1080/00927878608823308 |
| [4] | O. Chein and E.G. Goodaire, Loops whose Loop Rings in characteristic 2 are Alternative, Comm. in Algebra, 18(3) (1990), 659–688. · Zbl 0718.20034 · doi:10.1080/00927879008823938 |
| [5] | E.G. Goodaire, E. Jespers and C. Polcino Milies, ”Alternative Loop Rings”, North Holland Math. Series 184, Elsevier, Amsterdam, (1996). · Zbl 0878.17029 |
| [6] | E.G. Goodaire and M. M. Parmenter, Units in Alternative Loop Rings, Israel J. of Math. 53 (1986), 209–216. · Zbl 0594.17016 · doi:10.1007/BF02772859 |
| [7] | E.G. Goodaire and C. Polcino Milies, Isomorphism of Integral Loop Rings, Rend. Circ. Mat. Palermo XXXVII (1988), 126–135. · Zbl 0696.17015 · doi:10.1007/BF02844273 |
| [8] | E.G. Goodaire and C. Polcino Milies, Torsion Units in Alternative Loops, Proc. Amer. Math. Soc. 107 (1989) 7–15. · Zbl 0678.17021 · doi:10.1090/S0002-9939-1989-0953005-1 |
| [9] | E.G. Goodaire and C. Polcino Milies, The Torsion Product Property in Alternative Algebras, RT-MAT 95-15. · Zbl 0678.17021 |
| [10] | E.G. Goodaire and C. Polcino Milies, Finite Conjugacy in Alternative Loop Algebras, Comm. Algebra 24 (3), (1996), 881–889. · Zbl 0845.17024 · doi:10.1080/00927879608825608 |
| [11] | E.G. Goodaire and C. Polcino Milies, Ring Alternative Loops and their Loop Rings, Resenhas IME-USP 2 (1996), No. 1, 47–82. · Zbl 0860.17053 |
| [12] | E.G. Goodaire and C. Polcino Milies, On the Loop of Units of Alternative Loop Rings, Nova J. Alg. Geom. Game Theory, to appear. · Zbl 0868.20054 |
| [13] | E.G. Goodaire and C. Polcino Milies, Central Idempotents in Alternative Loop Algebras, preprint. |
| [14] | S.O. Juriaans, Trace Properties of Torsion Units in Group Rings II, preprint. |
| [15] | G. Leal and C. Polcino Milies, Isomorphic Group (and Loop) Algebras, J. Algebra 155 (1), (1993), 195–210. · Zbl 0773.20031 · doi:10.1006/jabr.1993.1039 |
| [16] | S.K. Sehgal, ”Units of Integral Group Rings”, Longman’s, Essex, 1993. · Zbl 0803.16022 |
| [17] | S.K. Sehgal, ”Topics in Group Rings”, Marcel Decker Inc. New York and Basel, 1978. · Zbl 0411.16004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
