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In this paper we describe all gradings by a finite abelian group G G on the following Lie algebras over an algebraically closed field F F of characteristic p ≠ 2 p\neq 2 : s l n ( F ) \mathfrak {sl}_n(F) ( n n not divisible by p p ), s o n ( F ) \mathfrak {so}_n(F) ( n ≥ 5 n\geq 5 , n ≠ 8 n\neq 8 ) and s p n ( F ) \mathfrak {sp}_n(F) ( n ≥ 6 n\geq 6 , n n even).

Assume that H is a finite-dimensional Hopf algebra over a field k and that A is an H-module algebra satisfying a polynomial identity (PI). We prove that if H is semisimple and A is H-semiprime, then A#H is semiprime. If H is cosemisimple, we show that the prime radical of A is H-stable.

In this paper we find necessary and sufficient conditions on a finite-dimensional Lie superalgebra under which any associative PI-envelope of L L is finite-dimensional. We also extend M. Scheunert’s result which enables one to pass from color Lie superalgebras to the ordinary ones, to the case of gradings by an arbitrary abelian group.

We prove that if A A is a Noetherian Z \mathbf Z -graded algebra, then the Jacobson radical of A A is nilpotent under mild hypotheses on A 0 A_0 . We also consider affine PI-algebras graded by torsion groups. Finally we prove a Nullstellensatz-type theorem for enveloping algebras of Lie color algebras.

We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C. This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element.

Recently Britten has proven an analog of Goldie’s theorem for the Jordan ring S S of symmetric elements in a ring with involution of characteristic not 2. In this paper we first extend Britten’s theorem to the situation where R R is an arbitrary ring and the Jordan ring is only an ample subring of the symmetric elements. We apply this result to show that if S S has ACC on quadratic ideals, then the (Jordan) nil radical of S S is nilpotent.

A well-known theorem of Jacobson asserts that a ring R R in which x n ( x ) = x {x^{n(x)}} = x for each x x in R R must be commutative. This paper gives a description of a ring with involution in which the above condition is imposed only on the symmetric elements. In particular, if R R is primitive, R R is either commutative or the 2 × 2 2 \times 2 matrices over a field, and, in general, any such R R is locally finite and satisfies a polynomial identity of degree 8.

The following theorem is proved: Let R be an algebra with involution over an uncountable field F. Then if the symmetric elements of R are algebraic, R is algebraic. In this paper we consider the following question: "Let R be an algebra with involution over a field F, and assume that the symmetric elements S of R are algebraic over F. Is R algebraic over F?" Previous results related to this question have been obtained by restricting the kind of algebraic relationships satisfied by the symmetric elements. For example, it was shown by Baxter and Martindale [1] for fields of characteristic not 2, and later by the author [5] for arbitrary fields, that if the symmetric elements are algebraic of bounded degree (or more generally, satisfy a polynomial identity), then R must be algebraic. Another such result concerns rings whose symmetric elements are periodic (that is, for each seS, there is some integer n(s)> 1 such that sn(s)=s). In this case, the author has shown [6], [7] that R must be algebraic; in fact R satisfies a polynomial identity. When R is a division ring, much more can be said: I. N. Herstein and the author [2] have shown that R must actually be commutative. Finally, it has been shown by Osborn [8] that if S is nil and F is uncountable, then R is nil. This answers for uncountable fields a question of McCrimmon [4, p. 391]: "If S is nil, is R nil?" An affirmative answer to this question in general would follow from an affirmative answer to the first question. For, as has been observed by both McCrimmon [4, p. 390] and Osborn [8, p. 306], if S is nil then R must be a radical ring. But if R is algebraic, every element of the radical is nil; thus R would be nil. The result presented here differs from those described above in that no additional restrictions are imposed on the symmetric elements. We prove: THEOREM. Let R be an algebra with involution over an uncountable field F. Then if the symmetric elements of R are algebraic, R is algebraic. Received by the editors January 8, 1971. AMS 1969 subject classifications. Primary 1658.

It has been shown by W. Baxter and W. S. Martindale that if R is an algebra with involution over a field F of characteristic not 2, and the symmetric elements of R are algebraic and satisfy a polynomial identity, then R is locally finite. This paper extends their result to an arbitrary field, giving a new proof which is independent of the characteristic of F. There are a number of results in the literature considering problems of the following sort: Given a ring with an involution such that the symmetric elements of the ring satisfy a certain property, under what conditions is the entire ring forced to satisfy the same property? One of the most striking such results is the theorem finally proved in its full generality by S. A. Amitsur [I]: If the symmetric elements of a ring with involution satisfy a polynomial identity, then the entire ring satisfies a polynomial identity. The result with which we are most concerned is a theorem of W. Baxter and W. S. Martindale [2] stating that if R is an algebra over a field F such that 1 E F, and the symmetric elements of R are algebraic and satisfy a polynomial identity, then R is locally finite. The purpose of this paper is to extend Baxter and Martindale's theorem to an arbitrary field F, giving a proof which is independent of the characteristic of F. It should be pointed out that it is impossible simply to generalize Baxter and Martindale's proof. For, when EE F, every element of R may be written as a sum of a symmetric and a skew element, and thus every element is a sum of algebraic elements. By a theorem of Procesi [4], R will then be locally finite if R satisfies a polynomial identity. Since this method will not work when F has characteristic 2, completely different methods are needed. Notice that this result generalizes a theorem of Kaplansky stating that an algebraic algebra satisfying a polynomial identity is locally finite [3, p. 167]. We will denote the involution on R by *; so * is an antiautomorphism of period 2. The symmetric elements of R, denoted by S, consist of all rER such that r* = r. R satisfies a polynomial identity Received by the editors May 13, 1970. AMS 1968 subiect classifications. Primary 1649, 1658.

We continue earlier work and compute the X-inner automorphisms of the ring of differential polynomials in one variable over an arbitrary domain. This is then applied to iterated Ore extensions. We also show that the ring of generic matrices has no nonidentity automorphisms which fix the center. In this paper we continue the study of X-inner automorphisms of filtered algebras begun in [7], which will be referred to as (I) in what follows. In (I) we described the X-inner automorphisms of the enveloping algebra of a Lie algebra, and of the ring of differential polynomials A = R[x; d] in one variable over a commutative domain R. Here we first show that the ring of generic matrices has only the trivial X-inner automorphism; equivalently, any automorphism fixing the center is the identity automorphism. We then extend our result from (I) on differential polynomials, allowing an arbitrary prime ring R as the coefficient ring. When R is a domain, we explicitly determine the group of X-inner automorphisms of A = R[x; d] in terms of R and d. We then apply these results to certain iterated Ore extensions; in particular, if the original coefficient ring is commutative, then the group of all X-inner automorphisms is abelian. In what follows, A will always denote a prime ring. Recall that a E Aut(A) is X-inner if it becomes inner when extended to the (right) Martindale quotient ring QO(A) of A; when A is an Ore domain, a is X-inner if and only if it becomes inner on the quotient division ring D of A. We shall need the following properties of QO(A): it is the (right) quotient ring of A with respect to the filter IF of all nonzero two-sided ideals (that is, QO(A) = lim ,HomA('A, A)), it is a prime ring with center C C(A) a field, called the extended center of A, and A may be imbedded in QO(A) as left multiplications. By construction, for any 0 #d x E A, there exists a nonzero I of A so that 0 =# xI c A. Fundamental in what follows is an internal characterization of X-inner automorphisms [8]. LEMMA 1. If a E Aut(A) is X-inner, say r' = s-'rs, all r E A for some s E QO(A), then there exist nonzero 'a, b E A such that sa = b and arb = b0r0a, all r E A. Conversely if arb = b0r0a, 'all r E A, for some nonzero a, b E A, then there exists s E QO(A) with sa b which induces a. Received by the editors April 23, 1982 and, in revised form, July 19, 1982. 1980 Mathematics Subject Classification. Primary 16A72, 16A38. ' Research supported in part by NSF grant No. MCS 81-01730. ? 1983 American Mathematical Society 0002-9939/82/0000-0756/$02.75