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Journal articles on the topic 'Units in rings and group rings'

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Published: 6 September 2023

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1

Jespers, Eric, and C. Polcino Milies. "Units of group rings." Journal of Pure and Applied Algebra 107, no. 2-3 (March 1996): 233–51. http://dx.doi.org/10.1016/0022-4049(95)00066-6.

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2

Kumari, P., M. Sahai, and R. K. Sharma. "Jordan regular units in rings and group rings." Ukrains’kyi Matematychnyi Zhurnal 75, no. 3 (April 11, 2023): 351–63. http://dx.doi.org/10.37863/umzh.v75i3.1130.

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UDC 512.5 The concept of Lie regular elements and Lie regular units was defined and studied by Kanwar, Sharma and Yadav in <em>Lie regular generators of general linear groups</em>, Comm. Algebra, <strong>40</strong>, № 4, 1304–1315 (2012)]. We introduce Jordan regular elements and Jordan regular units. It is proved that the order of the set of Jordan regular units in M ( 2 , Z 2 n ) is equal to a half of the order of U ( M ( 2 , Z 2 n ) ) . Further, we show that the group ring K G of a group G over a field K of characteristic 2 has no Jordan regular units.

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3

Bartholdi, Laurent. "On Gardam's and Murray's units in group rings." Algebra and Discrete Mathematics 35, no. 1 (2023): 22–29. http://dx.doi.org/10.12958/adm2053.

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We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.

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4

Farkas, Daniel R., and Peter A. Linnell. "Trivial Units in Group Rings." Canadian Mathematical Bulletin 43, no. 1 (March 1, 2000): 60–62. http://dx.doi.org/10.4153/cmb-2000-008-0.

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AbstractLet G be an arbitrary group and let U be a subgroup of the normalized units in ℤG. We show that if U contains G as a subgroup of finite index, then U = G. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

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5

Bist, V. "Torsion units in group rings." Publicacions Matemàtiques 36 (January 1, 1992): 47–50. http://dx.doi.org/10.5565/publmat_36192_04.

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6

Chatzidakis, Zoé, and Peter Pappas. "Units in Abelian Group Rings." Journal of the London Mathematical Society s2-44, no. 1 (August 1991): 9–23. http://dx.doi.org/10.1112/jlms/s2-44.1.9.

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7

Dekimpe, Karel. "Units in group rings of crystallographic groups." Fundamenta Mathematicae 179, no. 2 (2003): 169–78. http://dx.doi.org/10.4064/fm179-2-4.

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8

Herman, Allen, Yuanlin Li, and M. M. Parmenter. "Trivial Units for Group Rings with G-adapted Coefficient Rings." Canadian Mathematical Bulletin 48, no. 1 (March 1, 2005): 80–89. http://dx.doi.org/10.4153/cmb-2005-007-1.

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AbstractFor each finite group G for which the integral group ring ℤG has only trivial units, we give ring-theoretic conditions for a commutative ring R under which the group ring RG has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if R is a ring of finite characteristic and RG has only trivial units, then G has order at most 3.

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9

Herman, Allen, and Yuanlin Li. "Trivial units for group rings over rings of algebraic integers." Proceedings of the American Mathematical Society 134, no. 3 (July 18, 2005): 631–35. http://dx.doi.org/10.1090/s0002-9939-05-08018-4.

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10

Hoechsmann, K., and S. K. Sehgal. "Integral Group Rings Without Proper Units." Canadian Mathematical Bulletin 30, no. 1 (March 1, 1987): 36–42. http://dx.doi.org/10.4153/cmb-1987-005-6.

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AbstractIf A is an elementary abelian ρ-group and C one of its cyclic subgroups, the integral group rings ZA contains, of course, the ring ZC. It will be shown below, for A of rank 2 and ρ a regular prime, that every unit of ZA is a product of units of ZC, as C ranges over all cyclic subgroups.

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11

Juriaans, Stanley Orlando. "Torsion Units in Integral Group Rings." Canadian Mathematical Bulletin 38, no. 3 (September 1, 1995): 317–24. http://dx.doi.org/10.4153/cmb-1995-046-7.

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AbstractSpecial cases of Bovdi's conjecture are proved. In particular the conjecture is proved for supersolvable and Frobenius groups. We also prove that if is finite, α ∊ VℤG a torsion unit and m the smallest positive integer such that αm ∊ G then m divides .

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12

Valenti, Angela. "Torsion units in integral group rings." Proceedings of the American Mathematical Society 120, no. 1 (January 1, 1994): 1. http://dx.doi.org/10.1090/s0002-9939-1994-1186996-9.

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13

Parmenter, M. M. "UNITS AND ISOMORPHISM IN GROUP RINGS." Quaestiones Mathematicae 8, no. 1 (January 1985): 9–14. http://dx.doi.org/10.1080/16073606.1985.9631896.

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14

Lee, Gregory T. "Nilpotent Symmetric Units in Group Rings." Communications in Algebra 31, no. 2 (January 4, 2003): 581–608. http://dx.doi.org/10.1081/agb-120017331.

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15

DOOMS, ANN. "UNITARY UNITS IN INTEGRAL GROUP RINGS." Journal of Algebra and Its Applications 05, no. 01 (February 2006): 43–52. http://dx.doi.org/10.1142/s0219498806001569.

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16

Juriaans, Stanley Orlando. "Torsion units in integral group rings." Communications in Algebra 22, no. 12 (January 1994): 4905–13. http://dx.doi.org/10.1080/00927879408825111.

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17

Milies, Polcino C., and Sudarshan K. Sehgal. "Central units of integral group rings*." Communications in Algebra 27, no. 12 (January 1999): 6233–41. http://dx.doi.org/10.1080/00927879908826819.

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18

Hoechsmann, Klaus. "Constructing units in commutative group rings." Manuscripta Mathematica 75, no. 1 (December 1992): 5–23. http://dx.doi.org/10.1007/bf02567067.

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19

Li, Yuanlin, and M. M. Parmenter. "Hypercentral units in integral group rings." Proceedings of the American Mathematical Society 129, no. 8 (January 23, 2001): 2235–38. http://dx.doi.org/10.1090/s0002-9939-01-05848-8.

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20

Jespers, Eric, Gabriela Olteanu, Ángel del Río, and Inneke Van Gelder. "Central units of integral group rings." Proceedings of the American Mathematical Society 142, no. 7 (March 27, 2014): 2193–209. http://dx.doi.org/10.1090/s0002-9939-2014-11958-7.

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21

Luthar, I. S., and I. B. S. Passi. "Torsion units in matrix group rings." Communications in Algebra 20, no. 4 (January 1992): 1223–28. http://dx.doi.org/10.1080/00927879208824400.

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22

Danchev, Peter. "Idempotent Units of Commutative Group Rings." Communications in Algebra 38, no. 12 (December 15, 2010): 4649–54. http://dx.doi.org/10.1080/00927871003742842.

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23

Bovdi, Victor, and M. M. Parmenter. "Symmetric units in integral group rings." Publicationes Mathematicae Debrecen 50, no. 3-4 (April 1, 1997): 369–72. http://dx.doi.org/10.5486/pmd.1997.1853.

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24

Neisse, Olaf, and Sudarshan K. Sehgal. "Gauss Units in Integral Group Rings." Journal of Algebra 204, no. 2 (June 1998): 588–96. http://dx.doi.org/10.1006/jabr.1997.7379.

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25

Bovdi, A., Z. Marciniak, and S. K. Sehgal. "Torsion Units in Infinite Group Rings." Journal of Number Theory 47, no. 3 (June 1994): 284–99. http://dx.doi.org/10.1006/jnth.1994.1038.

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26

Giambruno, A., E. Jespers, and A. Valenti. "Group identities on units of rings." Archiv der Mathematik 63, no. 4 (October 1994): 291–96. http://dx.doi.org/10.1007/bf01189563.

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27

Riley, David M. "Group Rings With Hypercentral Unit Groups." Canadian Journal of Mathematics 43, no. 2 (April 1, 1991): 425–34. http://dx.doi.org/10.4153/cjm-1991-025-3.

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AbstractLet KG be the group ring of a group G over a field K and let U(KG) be its group of units. If K has characteristic p > 0 and G contains p-elements, then it is proved that U(KG) is hypercentral if and only if G is nilpotent and G′ is a finite p-group.

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28

Jespers, Eric, and G. Leal. "Units of integral group rings of hamiltonian groups*." Communications in Algebra 23, no. 2 (January 1995): 623–28. http://dx.doi.org/10.1080/00927879508825238.

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29

Nezhmetdinov, Timur I. "Groups of Units of Finite Commutative Group Rings*." Communications in Algebra 38, no. 12 (December 15, 2010): 4669–81. http://dx.doi.org/10.1080/00927870903451918.

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30

Herman, Allen, and Gurmail Singh. "On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes." Algebra 2014 (December 16, 2014): 1–5. http://dx.doi.org/10.1155/2014/842378.

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Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings.

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31

GONÇALVES, JAIRO Z., and ÁNGEL DEL RÍO. "A SURVEY ON FREE SUBGROUPS IN THE GROUP OF UNITS OF GROUP RINGS." Journal of Algebra and Its Applications 12, no. 06 (May 9, 2013): 1350004. http://dx.doi.org/10.1142/s0219498813500047.

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In this survey we revise the methods and results on the existence and construction of free groups of units in group rings, with special emphasis in integral group rings over finite groups and group algebras. We also survey results on constructions of free groups generated by elements which are either symmetric or unitary with respect to some involution and other results on which integral group rings have large subgroups which can be constructed with free subgroups and natural group operations.

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32

Goodaire, E. G., E. Jespers, and M. M. Parmenter. "Determining Units in Some Integral Group Rings." Canadian Mathematical Bulletin 33, no. 2 (June 1, 1990): 242–46. http://dx.doi.org/10.4153/cmb-1990-038-8.

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In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].

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33

Jespers, E. "Bicyclic Units in some Integral Group Rings." Canadian Mathematical Bulletin 38, no. 1 (March 1, 1995): 80–86. http://dx.doi.org/10.4153/cmb-1995-010-4.

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AbstractA description is given of the unit group for the two groups G = D12 and G = D8 × C2. In particular, it is shown that in both cases the bicyclic units generate a torsion-free normal complement. It follows that the Bass-cyclic units together with the bicyclic units generate a subgroup of finite index in for all n ≥ 3.

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34

Garcia, Vitor Araujo, and Raul Antonio Ferraz. "Central units in some integral group rings." Communications in Algebra 49, no. 9 (April 27, 2021): 4000–4015. http://dx.doi.org/10.1080/00927872.2021.1910284.

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35

Bien, M. H., M. Ramezan-Nassab, and D. H. Viet. "*-Group identities on units of division rings." Communications in Algebra 49, no. 7 (March 5, 2021): 3010–19. http://dx.doi.org/10.1080/00927872.2021.1887205.

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36

Ritter, J{ürgen, and Sudarshan K. Sehgal. "Integral group rings with trivial central units." Proceedings of the American Mathematical Society 108, no. 2 (February 1, 1990): 327. http://dx.doi.org/10.1090/s0002-9939-1990-0994785-7.

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37

Parmenter, M. M. "Conjugates of units in integral group rings." Communications in Algebra 23, no. 14 (January 1995): 5503–7. http://dx.doi.org/10.1080/00927879508825548.

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38

Johnson, F. E. A. "Arithmetic rigidity and units in group rings." Transactions of the American Mathematical Society 353, no. 11 (May 9, 2001): 4623–35. http://dx.doi.org/10.1090/s0002-9947-01-02816-1.

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39

Hoechsmann, K., and S. K. Sehgal. "Units in regular abelian p-group rings." Journal of Number Theory 30, no. 3 (November 1988): 375–81. http://dx.doi.org/10.1016/0022-314x(88)90009-1.

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40

Li, Yuanlin, and M. M. Parmenter. "Central units in integral group rings II." International Journal of Algebra 8 (2014): 47–55. http://dx.doi.org/10.12988/ija.2014.311106.

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41

Lee, Gregory T., and Ernesto Spinelli. "Group Rings Whose Symmetric Units are Solvable." Communications in Algebra 37, no. 5 (May 6, 2009): 1604–18. http://dx.doi.org/10.1080/00927870802116539.

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42

Ferraz, Raul Antonio, and Juan Jacobo Simón-Pınero. "Central Units in Metacyclic Integral Group Rings." Communications in Algebra 36, no. 10 (October 13, 2008): 3708–22. http://dx.doi.org/10.1080/00927870802158028.

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43

Lee, Gregory T., Sudarshan K. Sehgal, and Ernesto Spinelli. "Group rings whose unitary units are nilpotent." Journal of Algebra 410 (July 2014): 343–54. http://dx.doi.org/10.1016/j.jalgebra.2014.01.041.

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44

Hoechsmann, Klaus, and Jürgen Ritter. "Constructible units in abelian p-group rings." Journal of Pure and Applied Algebra 68, no. 3 (December 1990): 325–39. http://dx.doi.org/10.1016/0022-4049(90)90088-y.

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45

Oliver, Robert. "Central units in p-adic group rings." K-Theory 1, no. 5 (September 1987): 507–13. http://dx.doi.org/10.1007/bf00536982.

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46

Hoechsmann, Klaus, and Sudarshan K. Sehgal. "Units in regular elementary abelian group rings." Archiv der Mathematik 47, no. 5 (November 1986): 413–17. http://dx.doi.org/10.1007/bf01189981.

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47

Gonçalves, Daniel. "Simplicity of Partial Skew Group Rings of Abelian Groups." Canadian Mathematical Bulletin 57, no. 3 (September 1, 2014): 511–19. http://dx.doi.org/10.4153/cmb-2014-011-1.

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AbstractLet A be a ring with local units, E a set of local units for A, G an abelian group, and α a partial action of G by ideals of A that contain local units. We show that A*αG is simple if and only if A is G-simple and the center of the corner eδ0(A*αGe)eδ0 is a field for all e ∊ E. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

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48

Allen, P. J., and C. Hobby. "Units in Integral Group Rings of Some Metacyclic Groups." Canadian Mathematical Bulletin 30, no. 2 (June 1, 1987): 231–40. http://dx.doi.org/10.4153/cmb-1987-033-5.

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AbstractLet p be odd prime and suppose that G = 〈a, b〉 where ap-1 = bp = 1, a-1 ba = br, and r is a generator of the multiplicative group of integers mod p. An explicit characterization of the group of normalized units V of the group ring ZG is given in terms of a subgroup of GL(p - 1, Z). This characterization is used to exhibit a normal complement for G in V.

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49

Jespers, Eric, Guilherme Leal, and C. Polcino Milies. "Units of Integral Group Rings of Some Metacyclic Groups." Canadian Mathematical Bulletin 37, no. 2 (June 1, 1994): 228–37. http://dx.doi.org/10.4153/cmb-1994-034-0.

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AbstractIn this paper, we consider all metacyclic groups of the type 〈a,b | an - 1, b2 = 1, ba = aib〉 and give a concrete description of their rational group algebras. As a consequence we obtain, in a natural way, units which generate a subgroup of finite index in the full unit group, for almost all such groups.

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50

Dokuchaev, Michael A., and Sudarshan K. Sehgal. "Torsion units in integral group rings of solvable groups." Communications in Algebra 22, no. 12 (January 1994): 5005–20. http://dx.doi.org/10.1080/00927879408825118.

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