Two classes of rings generated by their units
Two classes of rings generated by their units
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DOI：
10.1016/0021-8693(74)90013-1
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发表时间：
1974-07
影响因子：
0.9
通讯作者：
M. Henriksen
中科院分区：
文献类型：
--
作者：
M. Henriksen
1953 年和 1954 年，K. Wolfson 和 D. Zelinsky 独立证明，特征不为 2 的除环上向量空间的所有线性变换的环的每个元素都是两个非奇异元素的和，参见 [16] 和 [17]。 1958 年，Skornyakov [15，第 14 页] 1671 提出了确定哪些规则环是由它们的单位生成的问题。 1969 年，虽然显然不知道 Skornyakov 的书，G. Ehrlich [3] 制作了一大类由其单位生成的规则环；也就是说，那些具有恒等式的环 R，其中 2 是一个单位，并且对于每个 a ER 都有一个单位 u ER，使得 aua= a。（另请参见[9]，作者在其中获得了此类规则环的其他特征。）最后，在[14]中，R. Raphael 启动了对其单位生成的环的系统研究，他将其称为 S 环。这篇笔记主要致力于概括拉斐尔的两个定理。他在[14]中表明，如果 R 是任何具有恒等式的环，并且 n> 1 是正整数，则环 R 的每个元素，以及具有来自 R 的条目的所有 n x n 矩阵的总和。在第 1 节中，我表明，在相同的假设下，R 的每个元素都是三个单位的和，并且我生成一类环 R，这样并非 R 的每个元素都是两个单位的和。如果 n> 1，则产生对于 R 的每个元素是两个单位之和的必要或充分条件的各种条件。Raphael 在 [14] 中还表明，如果 R 是一个具有同一性的环，使得对于每个 a E R 都有 ay E R 使得 aya= a 和 azy= ya2，并且如果 2 是 R 的一个单位，则 R 的每个元素都是四个单位的和。在第 2 节中，我证明，如果存在一个正整数 n，使得 (*) 对于每个 a ER，则存在一个 x ER，使得 axa= a 且 anx= xan，并且如果 max [2,(n-l)!] 是 R 的一个单位，那么 R 的每个元素都是有限数量单位的总和。满足(*)的原环R是相当特殊的，这种具有恒等性的环R的每个元素都是R的两个单元之和。
In 1953 and 1954, K. Wolfson and D. Zelinsky showed, independently, that every element of the ring of all linear transformations of a vector space over a division ring of characteristic not 2 is a sum of two nonsingular ones, see [16] and [17]. In 1958, Skornyakov [15, p. 1671 posed the problem of determining which regular rings are generated by their units. In 1969, while apparently unaware of Skornyakov’s book, G. Ehrlich [3] produced a large class of regular rings generated by their units; namely, those rings R with identity in which 2 is a unit and are such that for every a ER there is a unit u ER such that aua= a.(See also [9] where this author obtained other characterizations of such regular rings.) Finally, in [14], R. Raphael launched a systematic study of rings generated by their units, which he calls S-rings. This note is devoted mainly to generalizing two theorems of Raphael. He shows in [14] that if R is any ring with identity, and n> 1 is a positive integer, then every element of the ring R, of all n x n matrices with entries from R is a sum of 2n2 units. In Section 1 I show, under the same assumptions, that every element of R, is a sum of three units, and I produce a class of rings R such that not every element of R, is a sum of two units. A variety of conditions are produced that are either necessary or sufficient for every element of R, to be a sum of two units if n> 1. Raphael shows also in [14] that if R is a ring with identity such that for every a E R there is ay E R such that aya= a and azy= ya2, and if 2 is a unit of R, then every element of R is a sum of four units. In Section 2, I show that if there is a positive integer n such that (*) for every a ER, there is an x ER such that axa= a and anx= xan, and if max [2,(n-l)!] is a unit of R, then every element of R is a sum of a bounded number of units. Primitive rings R satisfying (*) are rather special and every element of such a ring R with identity is a sum of two units of R.