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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.

The following results are proved: IfR is a simple ring with unit, and for someaeR witha n in the center ofR, anyn, such that the centralizer ofa inR satisfies a polynomial identity of degreem, thenR satisfies the standard identity of degreenm. WhenR is not simple,R will satisfy a power of the same standard identity, provided thata andn are invertible inR. These theorems are then applied to show that ifG is a finite solvable group of automorphisms of a ringR, and the fixed points ofG inR satisfy a polynomial identity, thenR satisfies a polynomial identity, providedR has characteristic 0 or characteristicp wherep✗|G|.

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.

If R is an associative ring, we consider the special Jordan ring R + {R^ + } , and when R has an involution, the special Jordan ring S of symmetric elements. We first show that the prime radical of R equals the prime radical of R + {R^ + } , and that the prime radical of R intersected with S is the prime radical of S. Next we give an elementary characterization, in terms of the associative structure of R, of primeness of S. Finally, we show that a prime ideal of R intersected with S is a prime Jordan ideal of S.

In this paper the Lie structure of prime rings of characteristic 2 is discussed. Results on Lie ideals are obtained. These results are then applied to the group of units of the ring, and also to Lie ideals of the symmetric elements when the ring has an involution. This work extends recent results of I. N. Herstein, C. Lanski and T. S. Erickson on prime rings whose characteristic is not 2, and results of S. Montgomery on simple rings of characteristi c 2. 1* Prime rings* We first extend the results of Herstein [5]. Unless otherwise specified, all rings will be associative. If R is a ring, R has a Lie structure given by the product [x, y] = xy — yx, for x,yeR. A Lie ideal of R is any additive subgroup U of R with [u, r]e U for all u e U and reR. By a commutative Lie ideal we mean a Lie ideal which generates a commutative subring of R. Denote the center of R by Z. We recall that if R is prime, then the nonzero elements of Z are not zero divisors in R. In this case, if Z Φ 0 and F is the quotient field of R, then R ®ZF is a prime ring, every element of which can be written in the form r (x) α"1 for ae Z, a Φ 0. Thus R®ZF is naturally isomorphic to RZ~~\ the localization of R at Z. We will consider R imbedded in RZ~ι in the usual way (see [2]). We begin with some easy lemmas. LEMMA 1. If R is semi prime and U is a Lie ideal of R with u2 = 0 for all ue U, then U = 0.

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.