Direct sum cancellation of Noetherian modules (Q1383952)

Given a ring \(R\) with unit element and left \(R\)-modules \(A\), \(B\) and \(C\) that satisfy \(A\oplus C\simeq B\oplus C\), it is generally not true that \(A\simeq B\). Nevertheless, whenever \(C\) is noetherian, then \(A\) and \(B\) turn out to be indistinguishable by functions on the category of \(R\)-modules that respect short exact sequences. Specifically, it is shown that in this case \(A\) and \(B\) have isomorphic submodule series, that is, there are series of submodules \(0=A_0\subseteq A_1\subseteq\cdots\subseteq A_n=A\), \(0=B_0\subseteq B_1\subseteq\cdots\subseteq B_n=B\) and a permutation \(\sigma\), such that \(A_i/A_{i-1}\simeq B_{\sigma(i)}/B_{\sigma(i)-1}\) for \(i=1,2,\cdots,n\). In order to achieve this remarkable result, the author records the information that is needed about the category of all \(R\)-modules and its full subcategory of noetherian modules in a monoid, and he then uses theorems about monoids to prove the cancellation rules for modules.