Group Rings and Division Rings

Division algebras D with valuation v are studied, where D is lSnite-dimensional and totally ramified over its center F (i.e., the ramification index of v over wlF equals [D: F]). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When vlF is Henselian a full classification of the F-subalgebras of D is given. When F has a Henselian valuation v with separably closed residue field and A is any tame central simple F-algebra, an algorithm is given for computing the underlying division algebra of A from a suitable subgroup of A*/F*. Some examples are constructed using this valuation theory, including the first example of finitbdimensional F-central division algebras D1 and D2 with D1 XF D2 not a division ring, but D1 and D2 having no common subfield K D F. Valuation theory, long a basic tool in commutative algebra, has been relatively neglected in the study of division algebras, until quite recently. Nontheless, valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties, particularly algebras over iterated Laurent power series fields. For example, such division algebras have been key ingredients in Amitsur's noncrossed product construction [Am] and in Platonov's construction [P] of division algebras D with SK1(D) 7& 1. Valuations are not so prevalent on division algebras as on fields. But if a division algebra D does have a valuation, this structure contains a substantial amount of information about D which would scarcely be accessible otherwise. We consider here valued divisioll algebras D for which D is totQlly ramified and tame over its center F, i.e., for which the ramification index IrD rFI equals the dimension [D: F] of D over F and the characteristic char(D) does not divide [D: F]. (Here rD iS the value group of the valuation on D, and D is the residue division algebra. We assume throughout that [D: F] < oo.) Valued division algebras of this type appear, e.g., in Amitsur's noncrossed product paper [Am, §2], in Saltman's work on indecomposable division algebras [Sa], in certain of the MalcevNeumann division algebras considered by the first author and Amitsur [TA2,§4], etc. However, the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before. This may be because most past work on valued division algebras has concentrated on discrete valuations, when rD -Z; for such a valuation D is never totally ramified over its center F unless D = F (cf. (3.2) below). Received by the editors June 26, 1986. 1980 Mathematics Subject Clmsification (1985 Reon). Primary 16A39. 2Supported in part by F.N.R.S. 2Supported in part by the National Science Foundation. (:)1987 American Mathematlcal Society 0002-9947/87 $}.00 + $ 25 per page

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

If H is a maximal cone of a ring A such that the subring generated by H is a commutative integral domain that satisfies a certain centrality condition in A, then there exist a maximal cone H ′ H’ in a division ring A ′ A’ and an order preserving monomorphism of A into A ′ A’ , where the subring of A ′ A’ generated by H ′ H’ is a subfield over which A ′ A’ is algebraic. Hypotheses are strengthened so that the main theorems of the author’s earlier paper hold for maximal cones.

Nilpotent decomposition in integral group rings

In this paper, we present an isomorphism between the ring of general polynomials over a division algebra D with center F and the group ring of the free monoid with [D : F ] variables over D. Using this isomorphism, we define the characteristic polynomial of any matrix over any division algebra, i.e., a general polynomial with one variable over the algebra whose roots are precisely the left eigenvalues.Furthermore, we show how the left eigenvalues of a 4 × 4 quaternion matrices can be obtained by solving a general polynomial equation of degree 6.

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.

We study the existence of free subalgebras in division algebras, and prove the following general result: if \(A\) is a noetherian domain which is countably generated over an uncountable algebraically closed field \(k\) of characteristic \(0\), then either the quotient division algebra of \(A\) contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if \(k\) is an uncountable algebraically closed field and \(A\) is a finitely generated \(k\)-algebra that is a domain of GK-dimension strictly less than \(3\), then either \(A\) satisfies a polynomial identity, or the quotient division algebra of \(A\) contains a free \(k\)-algebra on two generators.

We study the zero divisor problem for group rings of some classes of wreath products. We prove also that some of these group rings embed into division rings or prime Artinian rings.

Let G be a locally indicable group, K a division ring, and KG a crossed-product group ring. In 1961, Ian Hughes proved that, up to KG-isomorphism, at most one division ring of fractions of KG satisfies a certain independence condition, now called Hughes freeness. This result was applied by others in work on division rings of fractions of group rings of free groups. In this article, we introduce concepts that illuminate Hughes' arguments, and we simplify the proof of the theorem.

So far our emphasis has been on studying rings in terms of the module categories they admit—that is, in terms of the representations of the rings as endomorphism rings of abelian groups. As we shall see the Wedderburn Theorem for simple artinian rings can be interpreted as asserting that a ring R is simple artinian if and only if the category R M is “the same” as the category D M for some division ring D. On the other hand, if D is a division ring, then the theory of duality from elementary linear algebra asserts that the categories D FM and FM D of finitely generated left D-vector spaces and right D-vector spaces are “duals” of one another.

It is shown that if F is a field of characteristic zero and G is a group such that the group ring F[G] is semilocal then G must be finite.A generalization to group rings over rings is given.A ring R is semilocal if R/J(R) is artinian, where I(R) denotes the Jacobson radical of R. R is said to be local if R/I(R) is a division ring.It is well known that the group ring is never local for a field of characteristic zero unless the group is trivial.As it is conjectured that J(F[G]) = (0) whenever Fis a field of characteristic zero, we expect F[G] semilocal to imply G finite, and this is easily proved if F is not algebraic over the rationals, by a theorem of Amitsur [1].The result in this paper may be interpreted as saying that the radical of the rational group ring cannot be "too large".We would like to thank D. S. Passman for his helpful suggestions.Lemma 1.Let K be a central subfield of a division ring D and let T G Mn(D), the full n by n matrix ring over D. Then the set S (T) = {k G K: 1 -k~ T is singular] has at most n elements.Proof.When written on the right, the elements of Mn(D) may be regarded as left Z)-linear transformations from the vector space D" to D".If 1 -k~] T is singular, there is a nonzero vector v G D" such that v(\ -k~ T) = 0, or vT = kv since k commutes with v. Hence k is an eigenvalue of T. Standard arguments of linear algebra show that eigenvectors in D" corresponding to distinct eigenvalues of T in the centre of D are Z)-linearly independent. Corollary.Let R be a completely reducible K-algebra and let x G R. Then the set S(x) = {k G K: 1 -k~ x is not a unit in R] is finite.The following clever lemma forms a major part of the proof of the theorem in [3],Lemma 2 (Formanek).Let K be a subfield of the reals and let x = 2"i aS

Even-order subgroups of finite-dimensional division rings

Finite subgroups occuring in finite-dimensional division algebras

We are given a division ring D with involution (*) and with a *-valuation V such that V(aa*b − baa*) >0 V(aa*b), for all nonzero elements a, b of D. We assume, further, that the associated residue class division ring D V is a (commutative) field with characteristic 0. In this work, we evidence two criteria in order for D to be either a standard quaternion division algebra or else a purely transcendental extension of its center. We apply one of these to answer our open Question 2.4.3 [2].

Let K/k be a separable field extension of degree 2, D be a finite-dimensional central division algebra over K with a K/k-involution τ, h be an Hermitian anisotropic form on a right D-vector space with respect to τ, and let U(h) be the unitary group of h. Then the reduced Whitehead group of its special linear subgroup is defined as follows: $$ \mathrm{SUK}_1^{\mathrm{an}}(h)={{{\mathrm{SU}(h)}} \left/ {{\left[ {U(h),U(h)} \right]}} \right.} $$ , where [U(h), U(h)] is the commutator subgroup of U(h). The first main result establishes a link between the above group and its analog SUK1(h) for the case of isotropic h (with respect to the same τ). Theorem. There exists a surjective homomorphism from $$ \mathrm{SUK}_1^{\mathrm{an}}(h) $$ to SUK1(h). Furthermore, we also give a solution of the conjugacy problem for special unitary subgroups of anisotropic Hermitian forms over quaternion division algebras as subgroups of their multiplicative groups. Bibliography: 32 titles.