## Publications

Group algebras with units satisfying a group identity ,
Proc. Amer. Math. Soc. 127(1999), no. 2, 327--336.
Abstract: Let K be a field and let G be a group. We prove a conjecture by Brian Hartley: "If G is torsion and units of the group algebra K[G] satisfy a group identity, then K[G] satisfies a polynomial identity."

Group algebras with units satisfying a group identity II, (with Donald S. Passman),
Proc. Amer. Math. Soc. 127 (1999), no. 2, 337--341.
Abstract: Let K be a field of characteristic p>0 and let G be a torsion group. We classify group algebras K[G] with units satisfying a group identity. This is the last in a series of papers, and the new results here concern the case of finite fields.

Group identities on units of locally finite algebras and twisted group algebras,
Communications in Algebra 28 (2000), no. 10, 4783--4802.
Abstract: We study locally finite algebras and twisted group algebras with units satisfying a group identity. As a preliminary result, we obtain a necessary condition for twisted group algebras to satisfy a generalized polynomial identity.

Some properties on rings with units satisfying a group identity,
Journal of Algebra 232 (2000), no. 1, 226--235.
Errata
Abstract: For convenience, a ring with units satisfying a group identity will be called a GI-ring. We show that GI-rings have the following properties which are also properites of PI-rings. (1)Any GI-ring is Dedekind finite(von Neumann finite). (2)Nilpotent elements of a semiprimitive GI-ring have bounded index. (3)The Kurosh problem has a positive answer for GI-algebras, namely, any algebraic GI-algebra is locally finite. We also study Hartley's problem for algebraic GI-algebras.

On nil subsemigroups of rings with group identities, (with K.I. Beidar and Wen-Fong Ke),
Communications in Algebra 30 (2002), 347--352.
Abstract: Let R be a unital ring satisfying a group identity. We prove that if B is a nil subsemigroup of R, then it is locally nilpotent, and B^d is contained in the sum of all nilpotent ideals of R, where the positive integer d is determined by the group identity. Note that the above result for PI-rings is due to Amitsur.

On units of twisted group algebras,
Journal of Algebra 250 (2002), no. 1, 271--282.
Abstract: We study units of twisted group algebras. Let G be a finite group and K be a field of characteristic p>0. Assume that K is not algebraic over a finite field. We find necessary and sufficient conditions for units of K^t[G] containing no nonabelian free subgroup. We also discuss what will happen when G is locally finite. For twisted group algebras of locally finite groups over any infinite field of characteristic p>0, we characterize those twisted group algebras with units satisfying a group identity. Finally, we include a new characterization for twisted group algebras to satisfy a polynomial identity.

Group identities and prime rings generated by units, (with Tsiu-Kwen Lee),
Communications in Algebra 31 (2003), 3305--3309.
Abstract: Let R be a prime ring generated by units. Suppose that units of R satisfy a group identity. We prove that R is either a domain or a full matrix ring over a finite field.

Automorphism groups of certain simple 2-(q,3,\lambda) designs constructed from finite fields , (with K.I. Beidar, W.-F. Ke and W.-R. Wu),
Finite fields and Their Applications 9 (2003), 400--412.

Prime Lie rings of derivations of commutative rings in characteristic 2, (with Donald S. Passman),
Journal of Algebra 311 (2007), no. 1, 352--364.
Abstract: Let R be a commutative associative ring with 1 and let Der(R) be the Lie ring of all derivations of R. Suppose that D is a Lie subring and an R-submodule of Der(R). When R is D-prime, we give necessary and sufficient conditions for D to be Lie prime. Since results of this nature are already known for rings R of characteristic different from 2, what is really new here is the characteristic 2 case.

Semiprime Lie rings of derivations of commutative rings,
Contemp. Math., 420, Amer. Math. Soc., 2006, 259--268.
Abstract: Let R be a commutative associative ring with 1 and let Der(R) be the Lie ring of all derivations of R. Suppose that D is a Lie subring and R-submodule of Der(R). If R is D-semiprime, we give necessary and sufficient conditions for D to be Lie semiprime.

Artinian Hopf algebras are finite dimensional, (with James J. Zhang),
Proc. Amer. Math. Soc. 135 (2007), no. 6, 1679-1680.
Abstract: We prove that an artinian Hopf algebra over a field is finite dimensional. This answers a question of Bergen.

Multiplicative Jordan decomposition in group rings of 3-groups, (with Donald S. Passman),
Journal of Algebra and its applications 8 (2009), no. 4, 509--519.
Abstract: In this paper, we essentially classify those finite 3-groups G having integral group rings with the multiplicative Jordan decomposition property. If G is abelian, then it is clear that Z[G] satisfies MJD. Thus, we are only concerned with the nonabelian case. Here we show that Z[G] has the MJD property for the two nonabelian groups of order 27. Furthermore, we show that there are at most three other specific nonabelian groups, all of order 81, with Z[G] having the MJD property. Unfortunately, we are unable to decide which, if any, of these three satisfies the appropriate condition.

Multiplicative Jordan decomposition in group rings of 2,3-groups, (with Donald S. Passman),
Journal of Algebra and its applications 9 (2010), no. 3, 483--492.
Abstract: In this paper, we essentially finish the classification of those finite 2,3-groups G having integral group rings with the multiplicative Jordan decomposition property. If G is abelian or a Hamiltonian 2-group, then it is clear that Z[G] satisfies MJD. Thus, we need only consider the nonabelian case. Recall that the 2-groups with MJD were completely determined by Hales, Passi and Wilson, while the corresponding 3-groups were almost completely determined by the present authors. Thus, we are concerned here, for the most part, with groups whose order is divisible by 6. As it turns out, there are precisely three nonabelian 2,3-groups, of order divisible by 6, with Z[G] satisfying MJD. These have orders 6, 12, and 24. In view of another result of Hales, Passi and Wilson, this essentially completes the classification of all finite groups with MJD.

Multiplicative Jordan decomposition in group rings and p-groups with all noncyclic subgroups normal,
Journal of Algebra 371 (2012), 300--313.
Abstract: Let p be a prime and let G be a finite p-group. We show that if the integral group ring Z[G] satisfies the multiplicative Jordan decomposition property, then every noncyclic subgroup of G is normal. This is used to simplify the work of Hales, Passi and Wilson on the classification of integral group rings of finite 2-groups with the multiplicative Jordan decomposition property.

Multiplicative Jordan decomposition in group rings of 3-groups, II, (with Donald S. Passman),
Communications in Algebra 42 (2014), 2633--2639.
Abstract: In this paper we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that Z[G] satisfies MJD. In the nonabelian case, we show that Z[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 27.

Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3, (with Donald S. Passman),
Journal of Algebra 388 (2013), 203--218.
Abstract: If G is a finite group whose integral group ring Z[G] has the multiplicative Jordan decomposition property, then it is known that all Wedderburn components of the rational group ring Q[G] have degree at most 3. While degree 3 components can occur, we prove here that if they do, then certain central units in Z[G] cannot exist. With this, we are able to greatly simplify the argument that characterizes those 3-groups with integral group ring having MJD. Furthermore, we show that if G is a nonabelian semidirect product of a cyclic group of prime order p >7 with a cyclic 3-group, then Z[G] does not have MJD.

Groups with certain normality conditions, (with Donald S. Passman),
Communications in Algebra 44 (2016), 3308--3323.
Abstract: We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.