基本子群的交换子：curiouser和curiouser
Commutators of elementary subgroups: curiouser and curiouser
摘要：
设$R$是任意具有$1$，$n\ge 3$的结合环，$A,B$是$R$的双边理想.在我们以前与Roophih Hazrat [17，15]的联合工作中，我们发现了混合换位子群$[E(n,R,A),E(n,R,B)]$的生成集。后来在[29，34]中，我们注意到我们以前的结果可以大大改进，并且$[E(n,R,A),E(n,R,B)]$是由1）基本共轭$z_{ij}(ab,c)=t_{ij}(c)t_{ji}(ab)t_{ij}(-c)$和$z_{ij}(ba,c)$，2）基本共轭子$[t_{ij}(a),t_{ji}(b)]$生成的，其中$1\le i\neq j\le n$，$a\in A$，$b\in B$，$c\in R$。后来在[33，35]中我们注意到，对于第二种类型的生成器，甚至只需固定一对索引$(i,j)$。在这里，我们在另一个完全出乎意料的方向改进了上述结果，并证明了$[E(n,R,A),E(n,R,B)]$是由初等分解子$[t_{ij}(a),t_{hk}(b)]$单独生成的，其中$1\le i\neq j\le n$，$1\le h\neq k\le n$，$a\in A$，$b\in B$。这使我们能够修改技术的相对本地化，特别是，给非常短的证明一些最近的结果，如产生部分相对化的基本群体$E(n,A)^{E(n,B)}$，%%正常的$E(n,AB+BA)$内$[E(n,R,A),E(n,R,B)]$，多个换向器公式，换向器宽度，等等。
英文摘要：
Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set for the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$. Later in [29,34] we noticed that our previous results can be drastically improved and that $[E(n,R,A),E(n,R,B)]$ is generated by 1) the elementary conjugates $z_{ij}(ab,c)=t_{ij}(c)t_{ji}(ab)t_{ij}(-c)$ and $z_{ij}(ba,c)$, 2) the elementary commutators $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in R$. Later in [33,35] we noticed that for the second type of generators, it even suffices to fix one pair of indices $(i,j)$. Here we improve the above result in yet another completely unexpected direction and prove that $[E(n,R,A),E(n,R,B)]$ is generated by the elementary commutators $[t_{ij}(a),t_{hk}(b)]$ alone, where $1\le i\neq j\le n$, $1\le h\neq k\le n$, $a\in A$, $b\in B$. This allows us to revise the technology of relative localisation, and, in particular, to give very short proofs for a number of recent results, such as the generation of partially relativised elementary groups $E(n,A)^{E(n,B)}$, %% normality of $E(n,AB+BA)$ inside $[E(n,R,A),E(n,R,B)]$, multiple commutator formulas, commutator width, and the like.
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