S. A. Amitsur

On central division algebras

For any integern such that 8|n or for which there exists an odd primeq such thatq2|n, there is a central division algebra of dimensionn2 over its center which is not a crossed product. The algebra constructed in this paper is the algebraQ(X1,…,X)m, the algebra generated over the rationalQ bym(≧2) generic matrices.

Identities in rings with involutions

A ringR with an involutiona →a* which satisfies a polynomial identityp[x1,…,xd;x*1, …,x*d]=0 satisfies- an identity which does not include thex*. This generalizes the result of [1] where the symmetric elements ofR were assumed to satisfy an identity.

Division algebras of degree 4 and 8 with involution

We develop necessary and sufficient conditions for central simple algebras to have involutions of the first kind, and to be tensor products of quaternion subalgebras. The theory is then applied to give an example of a division algebra of degree 8 with involution (of the first kind), without quaternion subalgebras, answering an old question of Albert; another example is a division algebra of degree 4 with involution (*) has no (*)-invariant quaternion subalgebras.

Rational identities and applications to algebra and geometry

Let 1) be a division algebra with a center F containing a field C. Intuitively speaking a rational identity p[x, y. ...I = 0 means a relation which holds identically in T1 and is formed from a finite number of noncommutative indeterminates x, y, 2, ... by addition, multiplication, and taking inverses, whenever those exist. It is the last step which makes the crucial ditference between polynomial identities and rational idelltities. In contrast with polynomial identities, where the existence of such a relation implies finiteness over the center, one encounters rational identities which hold for all division algebra.. 5 Some of these arc very useful. The fundamental theorem of projective geometry follows easily by using the identitv s [x 1 + (y ’ x3-‘]-’ xvx 0 ([.5]). The identity

Rings with involution

Structure theorems for ringsR with involution whose symmetric elements satisfy a polynomial identity are obtained. In particular, it is shown that such rings satisfy polynomial identities.

PI-algebras and their cocharacters

Abstract All PI-algebras R satisfy identities of the form: f ∗ [x, y] = Σ α σ x σ(1) y 1 x σ (2)y 2 … y n − 1 x σ(n) (Theorem A.) The existence of these identities imply also that cocharacters x n ( R ) lie in a hook-shaped strip of width depending on the degree of the minimal identity of R (Theorem C). This extends a characterization of rings satisfying a Capelli identity (Theorem B).

Kummer subfields of Malcev-Neumann division algebras

The abelian Galois subfields of Malcev-Neumann formal series division rings are determined. The results obtained in this paper lead to a lower bound for the rank of Galois splitting fields of universal division algebras.

Polynomials over division rings

LetD be a division ring with a centerC, andD[X1, …,XN] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsMn(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X1, …,Xm]-modules are finite-dimensionalD-spaces.

Identities and linear dependence

Central polynomial identities are used to construct alternating central identities by which new identities are obtained. These identities express the linear dependence ofn2+1 generic matrices, and so yield slight generalizations and simplified proofs of a result of Formanek, the theorem about Azumaya algebras of M. Artin and a recent result of Cauchon.

Elements of reduced trace 0

Every elementr of reduced trace 0 in a simple finite dimensional algebraR is a sum of at most 2 commutators. IfR innot a division ring thenr is a commutator, unlessr is a scalar (in which case char(R)≠0). The method of proof provides a generic division algebra of transcendence degreen2−1.