The Burnside problem
The Burnside problem
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DOI：
10.1016/0021-8693(66)90031-7
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发表时间：
1966-11
影响因子：
0.9
通讯作者：
C. Procesi
中科院分区：
文献类型：
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作者：
C. Procesi
一般的伯恩赛德问题最近已解决的负面Golod和Safarevic [I]。人们仍然可以尝试在扭群G上找到使其局部有限的充分条件。这在定理2中得以实现：如果G可以嵌入萨蒂斯多项式恒等式的环R中，则G是局部Jinite.显然这个条件是不必要的，因为问题是局部问题，而R上的恒等式是全局的.这没有什么害处;人们可以稍微改变这个条件，使它成为必要和充分的。我们以这种方式进行：我们说一个环R局部满足恒等式，如果R的每一个生成子环都满足某个恒等式（取决于子环）。若G是局部有限扭群，则任意域F上的群代数F（G）局部满足恒等式，这是很容易证明的。反之，若G嵌入在局部满足恒等式的环R中，则G的一个n-生成子群H嵌入在R的一个n-生成子环中，该n-生成子环通过假设满足一个恒等式;定理2则适用并确保H是有限的。在定理2的陈述中，我们必须小心在R中成立的恒等式的种类，因为R不被假定为域上的代数，并且我们希望避免平凡恒等式。设Q是R上的算子环，p = Cwtijxi/xi，qij EQ，是R中的恒等式，我们假设wtijx= 0，x ∈ R，对所有wcij，x= 0.这个假设是很自然的;如果不满足这个假设，恒等式p在R的某个非零子环上是绝对平凡的，我们就不能指望得到关于R的这一部分的实质性信息。上述假设保证了R的诣零根N是局部幂零的，并且R/N满足系数为-&1的恒等式。在特殊情况下，当R是一个代数域，定理2是一个推论，我们的第一个定理是独立的利益，完全是环理论。这个定理证明了满足多项式恒等式的域F上的代数R是代数的，如果它有一个由代数元素组成的基。
The general Burnside problem has been recently settled in the negative by Golod and Safarevic [I]. One can still try to find sufficient conditions on a torsion group G to make it locally finite. This is accomplished in Theorem 2: If G can be embedded in a ring R which satis $ es a polynomial identity, then G is locally Jinite.Clearly this condition is not necessary, because the problem is a local one while the identity on R is global. This is of little harm; one can change the condition slightly to make it necessary and sufficient. We proceed in this way: We say that a ring R satisfies identities locally if every finitely generated subring of R satisfies some identity (depending on the subring). If G is a locally finite torsion group then the group algebra F (G) over any field F satisfies identities locally, as can be easily verified. Conversely, if Gis embedded in a ring R satisfying identities locally, a finitely generated subgroup H of G is embedded in a finitely generated subring of R which by hypothesis satisfies an identity; Theorem 2 then applies and ensures that H is finite. In the statement of Theorem 2 we have to be careful about the kind of identities holding in R, because R is not assumed to be an algebra over a field and we want to avoid trivial identities. If Q is a ring of operators on R andp= Cwtijxi/xi,, qij EQ, is the identity holding in R, we assume that wtijx= 0, x E R, for all wcij implies x= 0. This assumption is a natural one; if it is not satisfied the identity p is absolutely trivial on some nonzero subring of R and we cannot hope to have substantial information on this piece of R. The hypothesis mentioned above ensures that the nil radical N of R is locally nilpotent and R/N satisfies an identity with coefficients-& 1. In the special case when R is an algebra over a field, Theorem 2 is a corollary of our first theorem which is of independent interest and is entirely ring-theoretical. This theorem affirms that an algebra R over a field F satisfying a polynomial identity is algebraic if it has a basis consisting of algebraic elements.