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We construct, for any prime p p , a non-cyclic central simple algebra of degree p 2 p^2 with p 2 p^2 -central elements. This construction answers a problem of Peter Roquette.

Given a quaternion division algebra D, a noncentral element e ∈ D × is called pure if its square belongs to the center. A theorem of Rowen and Segev (2004) asserts that for any quaternion division algebra D of positive characteristic > 2 and any pure element e ∈ D × the quotient D × /X(e) of D × by the normal subgroup X(e) generated by e, is abelian-by-nilpotent-by-abelian. In this note we construct a quaternion division algebra D of characteristic zero containing a pure element e ∈ D such that D × /X(e) contains a nonabelian free group. This demonstrates that the situation in characteristic zero is very different.

We examine some properties of bicyclic algebras, i.e. the tensor product of two cyclic algebras, defined over a purely transcendental function field in one variable. We focus on the following problem: When does the set of local invariants of such an algebra coincide with the set of local invariants of some cyclic algebra? Although we show this is not always the case, we determine when it happens for the case where all degeneration points are defined over the ground field. Our main tool is Faddeev's theory. We also study a geometric counterpart of this problem (pencils of Severi-Brauer varieties with prescribed degeneration data).

Let k be an algebraically closed field of characteristic 0. We prove that any division algebra over k(x, y) whose ramification locus lies on a quartic curve is cyclic.

Using elementary methods we prove a theorem of Rost, Serre, and Tignol that any division algebra of degree 4 over a C 3 C_{3} -field containing − 1 \sqrt {-1} is cyclic. Our methods also show any division algebra of degree 8 over a C 2 C_{2} -field containing − 1 4 \sqrt [4 ]{-1} is cyclic.

It is shown that the Brauer factor set ( c i j k ) ({c_{ijk}}) of a finite-dimensional division algebra of odd degree n n can be chosen such that c i j i = c i i j = c j i i = 1 {c_{iji}} = {c_{iij}} = {c_{jii}} = 1 for all i , j i,j and c i j k = c k j i − 1 {c_{ijk}} = c_{kji}^{ - 1} . This implies at once the existence of an element a ≠ 0 a \ne 0 with tr ( a ) = tr ( a 2 ) = 0 {\text {tr}}(a) = {\text {tr}}({a^2}) = 0 ; the coefficients of x n − 1 {x^{n - 1}} and x n − 2 {x^{n - 2}} in the characteristic polynomial of a a are thus 0 0 . Also one gets a generic division algebra of degree n n whose center has transcendence degree n + ( n − 1 ) ( n − 2 ) / 2 n + (n - 1)(n - 2)/2 , as well as a new (simpler) algebra of generic matrices. Equations are given to determine the cyclicity of these algebras, but they may not be tractable.

Various algebraic reductions are made to facilitate computer verification of the following result: If x and y are 8 × 8 8 \times 8 matrices such that [x, y] is regular, tr ⁡ ( x ) = 0 \operatorname {tr} (x) = 0 , and, with respect to the canonical symplectic involution, x is symmetric and y is antisymmetric, then the element ( x + [ x , y ] x [ x , y ] − 1 ) 2 {(x + [x,y]x{[x,y]^{ - 1}})^2} satisfies a minimal equation of degree ⩽ 2 \leqslant 2 .

Results of Bjork and Sabbagh are extended and generalized to provide a Krull-Schmidt theory over a general class of semiperfect rings which includes left perfect rings, right perfect rings, and semiperfect PI-rings whose Jacobson radicals are nil. The object of this paper is to lay elementary foundations to the study of f.g. (i.e. finitely generated) modules over rings which are almost Artinian, with the main goal being a theory following the lines of the Azumaya-Krull-Remak-SchmidtWedderburn theorem (commonly called Krull-Schmidt); in other words we wish to show that a given f.g. module is a finite direct sum of indecomposable submodules whose endomorphism rings are local. Previous efforts in this direction include [2, 3, 4, 6], and in particular the results here extend some results of [2, 4, 6]. The focus here will be on a "Fitting's lemma" approach applied to semiperfect rings, cf. Theorem 8. We recall the definition from [1], which will be used as a standard reference. R is semiperfect if its Jacobson radical J is idempotent-lifting and R/J is semisimple Artinian; equivalently every f.g. module M has a projective cover (an epic map ir: P -+ M, where P is projective and ker ir is a small submodule of P). Projective covers are unique up to isomorphism by [1, Lemma 17.17]. In what follows, module means "left module". PROPOSITION 1. If R is semiperfect, then every f.g. module M is a finite direct sum of indecomposable submodules. PROOF. Let ir: P -* M be a projective cover. Then P has an indecomposable decomposition of some length (cf. [1, Theorem 27.12]) and we show by induction on t that M also has an indecomposable decomposition of length < t. Indeed this is tautological if M is indecomposable, so assume M = M1 0 M2. By [1, Lemma 17.17] there are projective covers iri: Pi -? Mi, where Pi are direct summands of P, and in fact P1 0 P2 . P by [1, Exercise 15.1], so we can proceed inductively on M1 and M2. Q.E.D. REMARK 2. By [1, Theorem 27.6] an f.g. R-module M is a direct sum of (indecomposable) modules having local endomorphism iff EndRM is semiperfect, so we ask: For which modules M is EndRM semiperfect? (This is why it is natural to study semiperfectrings R.) In [4, Example 2.1] Bjork found an example of a cyclic module M = R/L over a semiprimary ring R such that E = EndRM is not Received by the editors January 4, 1984 and, in revised form, December 18, 1984 and May 17, 1985. 1980 Mathematics Subject Classification. Primary 16A51, 16A65; Secondary 16A50, 16A64.

Let ( R , ∗ ) (R, \ast ) denote a ring R R with involution ( ∗ ) ( \ast ) , where “involution” means “ anti - automorphism of order ≦ two {\text {anti - automorphism of order}} \leqq {\text { two}} ". We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of ( R , ∗ ) (R, \ast ) is an ideal of R R stable under ( ∗ ) ( \ast ) , and the center of ( R , ∗ ) (R, \ast ) is the set of central elements of R R which are fixed under ( ∗ ) ( \ast ) . Then we say ( R , ∗ ) (R, \ast ) is prime when the product of any two nonzero ideals of ( R , ∗ ) (R, \ast ) is nonzero; similarly ( R , ∗ ) (R, \ast ) is semiprime when any power of a nonzero ideal of ( R , ∗ ) (R, \ast ) is nonzero. The main result of this paper is a strong analogue to Posner’s theorem [5], namely that any prime ( R , ∗ ) (R, \ast ) with polynomial identity has a ring of quotients R T {R_T} , formed merely by adjoining inverses of nonzero elements of the center of ( R , ∗ ) (R, \ast ) . This quotient ring ( R T , ∗ ) ({R_T}, \ast ) is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.

J. Fisher [3] initiated the study of maximal quotient rings of semiprime PI-rings by noting that the singular ideal of any semiprime Pi-ring R is 0; hence there is a von Neumann regular maximal quotient ring Q ( R ) Q(R) of R. In this paper we characterize Q ( R ) Q(R) in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of Q ( R ) Q(R) to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of Q ( R ) Q(R) are given, and Q ( R ) Q(R) turns out to have an involution when R has an involution.

Levy [21 has examined semiprime rings which are irredundant subdirect products of prime rings. In this note we look at the role of inessential prime ideals and see how every semiprime ring is a subdirect product of (i) a semiprime ring which is an irredundant subdirect product of prime rings, and (ii) a semiprime (nonprime) ring, all of whose prime ideals are essential. This leads to a direct sum decomposition of maximal left quotient rings of semiprime rings with left singular ideal zero. Let R be a semiprime ring, i.e., R has no nonzero nilpotent ideals. Let ?P -prime ideals of RI. It is well known that niP e St 0. If A is a subset of R then let Ann A = lr e Rlar = 0 for all a e Al, and let Ann'A = tr e RIra = 0 for all a e Al. Since R is semiprime, Ann A = Ann'A for any ideal A of R. (Proof. (A Ann'A)2 C (Ann'A)A 0, so A Ann'A = 0, implying Ann'A C Ann A; Ann A C Ann'A follows by symmetry.) We say an ideal A is essential if A n B 4 0 for any nonzero ideal B of R. (Note. 0 is an inessential ideal.) This is equivalent to Ann A = 0 by an argument similar to the previous one. We say an ideal B of R is semiprime if R/B is a semiprime ring. Lemma 1. If B is an ideal of R then Ann B is a semiprime ideal of R. In fact, Ann B =nfIP E 9TB Pi. Proof. B C nflPE TPjB C P1. Now let B' = nfP e YIB t Pi. BB' C B r B'= 0 so B C Ann B. On the other hand, for any P B, B Ann B=0 C P, Received by the editors August 3, 1973. AMS (MOS) subject classifications (1970). Primary 16A08, 16A12.

Let M n ( F ) {M_n}(F) be an n × n n \times n matrix ring with entries in the field F, and let S k ( X 1 , … , X k ) {S_k}({X_1}, \ldots ,{X_k}) be the standard polynomial in k variables. Amitsur-Levitzki have shown that S 2 n ( X 1 , … , X 2 n ) {S_{2n}}({X_1}, \ldots ,{X_{2n}}) vanishes for all specializations of X 1 , … , X 2 n {X_1}, \ldots ,{X_{2n}} to elements of M n ( F ) {M_n}(F) . Now, with respect to the transpose, let M n − ( F ) M_n^ - (F) be the set of antisymmetric elements and let M n + ( F ) M_n^ + (F) be the set of symmetric elements. Kostant has shown using Lie group theory that for n even S 2 n − 2 ( X 1 , … , X 2 n − 2 ) {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}}) vanishes for all specializations of X 1 , … , X 2 n − 2 {X_1}, \ldots ,{X_{2n - 2}} to elements of M n − ( F ) M_n^ - (F) . By strictly elementary methods we have obtained the following strengthening of Kostant’s theorem: S 2 n − 2 ( X 1 , … , X 2 n − 2 ) {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}}) vanishes for all specializations of X 1 , … , X 2 n − 2 {X_1}, \ldots ,{X_{2n - 2}} to elements of M n − ( F ) M_n^ - (F) , for all n. S 2 n − 1 ( X 1 , … , X 2 n − 1 ) {S_{2n - 1}}({X_1}, \ldots ,{X_{2n - 1}}) vanishes for all specializations of X 1 , … , X 2 n − 2 {X_1}, \ldots ,{X_{2n - 2}} to elements of M n − ( F ) M_n^ - (F) and of X 2 n − 1 {X_{2n - 1}} to an element of M n + ( F ) M_n^ + (F) , for all n. S 2 n − 2 ( X 1 , … , X 2 n − 2 ) {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}}) vanishes for all specializations of X 1 , … , X 2 n − 3 {X_1}, \ldots ,{X_{2n - 3}} to elements of M n − ( F ) M_n^ - (F) and of X 2 n − 2 {X_{2n - 2}} to an element of M n + ( F ) M_n^ + (F) , for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.

Any central simple algebra of degree n n split by a Galois extension with dihedral Galois group of degree 2 n 2n is, in fact, a cyclic algebra. We assume that the centers of these algebras contain a primitive n n th root of unity.