论文标题
有限生成的模块的同等产生向量上的通勤环
Equivalent generating vectors of finitely generated modules over commutative rings
论文作者
论文摘要
令$ r $为具有身份的可交换戒指，让$ m $为$ r $ $模块，由$μ$元素生成，但不是更少。我们用$ \ operatotorName {sl} _n（r）$ $ n \ times n $矩阵的组超过$ r $，带有确定性$ 1 $。我们用$ \ operatorname {e} _n（r）$ $ \ permatatorName {sl} _n（r）$的子组由矩阵生成的矩阵产生，与身份不同，这是由单个偏高系数与身份不同的。给定$ n \geμ$和$ g \ in \ weft \ {\ operatorName {sl} _n（r），\ propatatorName {e} _n（r）\ right \} $，我们研究$ g $的$ g $的动作。组件产生$ m $。假设$ m $有限地呈现，并且$ r $是基本的分隔环或几乎本地的全球连贯的prüfer环，我们获得了$ \ peripatatorName {v} _n（m）/g $的描述，它扩展了作者早些时候在Quasi-e-ealclelidean ring上的有限生成的模块。
Let $R$ be a commutative ring with identity and let $M$ be an $R$-module which is generated by $μ$ elements but not fewer. We denote by $\operatorname{SL}_n(R)$ the group of the $n \times n$ matrices over $R$ with determinant $1$. We denote by $\operatorname{E}_n(R)$ the subgroup of $\operatorname{SL}_n(R)$ generated by the the matrices which differ from the identity by a single off-diagonal coefficient. Given $n \ge μ$ and $G \in \left\{\operatorname{SL}_n(R),\operatorname{E}_n(R)\right\}$, we study the action of $G$ by matrix right-multiplication on $\operatorname{V}_n(M)$, the set of elements of $M^n$ whose components generate $M$. Assuming that $M$ is finitely presented and that $R$ is an elementary divisor ring or an almost local-global coherent Prüfer ring, we obtain a description of $\operatorname{V}_n(M)/G$ which extends the author's earlier result on finitely generated modules over quasi-Euclidean rings.
