Gerstenhaber showed in 1961 that any commuting pair of n x n matrices over afield k generates a k-algebra A of k-dimension \leq n. A well-known exampleshows that the corresponding statement for 4 matrices is false. The questionfor 3 matrices is open. Gerstenhaber's result can be looked at as a statement about the relationbetween the length of a 2-generator finite-dimensional commutative k-algebra Aand the lengths of faithful A-modules. Wadsworth generalized this result to alarger class of commutative rings than those generated by two elements over afield. We recover his generalization, using a slightly improved argument. We then explore some examples, raise further questions, and make a bit ofprogress toward answering some of them. An appendix gives some results on generation and subdirect decompositions ofmodules over not necessarily commutative Artinian rings, generalizing a specialcase used in the paper. What I originally thought of as my main result turned out to have beenanticipated by Wadsworth, so I probably won't submit this for publicationunless I find further strong results to add. However, others may findinteresting the observations, partial results, and questions noted, and perhapsmake some progress on them.
展开▼
机译：Gerstenhaber于1961年证明，在场k上任何对换的n x n矩阵对都会生成k维\ leq n的k代数A。一个著名的例子表明，对应于4个矩阵的语句为false。 3个矩阵的问题已打开。 Gerstenhaber的结果可以看作是关于2发生器有限维可交换k代数A的长度与忠实A模块的长度之间的关系的陈述。 Wadsworth将这一结果推广到比在场上的两个元素所产生的交换环更大的一类交换环。我们使用稍微改进的论点恢复了他的概括。然后，我们探索一些示例，提出进一步的问题，并在回答其中一些方面取得一些进展。附录给出了在不一定是可交换的Artinian环上模块的生成和子分解的一些结果，并概括了本文中使用的一种特殊情况。我最初认为的主要结果原来是Wadsworth所期望的，因此除非找到进一步的强劲结果，否则我可能不会将其提交出版。但是，其他人可能会发现有趣的观察结果，部分结果和指出的问题，并可能在此基础上取得一些进展。
展开▼
