Krull dimension of bimodules. (Q876360)

Let \(R\) and \(S\) be unital rings with identity, and let \(M\) be a left \(S\)-, right \(R\)-bimodule. One says that the bimodule \(_SM_R\) is `Krull symmetric' if the modules \(_SM\) and \(M_R\) both have Krull dimension, denoted by \(\text{k}(_SM)\) and \(\text{k}(M_R )\), respectively, and \(\text{k}(_S(M/N))=\text{k}((M/N)_R)\) for all sub-bimodules \(N\) of \(_SM_R\). The authors prove that if \(_SM_R\) is a Krull symmetric bimodule such that the module \(_SM\) is finitely generated then \(\text{k}(M_R)=\text{k}((M/K)_R)\) for some prime submodule \(K\) of \(M_R\) such that \((M/K)_R\) is k-critical. The authors also prove the following bimodule analogue of a result from 1973 of Lambek and Michler: if \(_SM_R\) is a Noetherian bimodule (this means that \(M\) is Noetherian both as a left \(S\)-module and as a right \(R\)-module), then the right \(R\)-module \(M\) is Artinian if and only if every irreducible prime submodule of \(M_R\) is maximal.