Some remarks on the Chinese remainder theorem (Q1117962)

Let A be a module over any ring and let \({\mathcal F}=\{A_ 1,...,A_ m\}\) be a family of submodules. Then the author considers the natural embedding \(\phi =\phi ({\mathcal F})\) of \(A/(A_ 1\cap... \cap A_ n)\) into A(\({\mathcal F}):=\{(a_ i)\in \prod^{n}_{i=1}A/A_ i|\) \(a_ i\equiv a_ j mod(A_ i+A_ j)\), all i,j\(\}\) and says that the generalized Chinese Remainder Theorem (GCRT) holds for \({\mathcal F}\) if \(\phi\) is onto. He considers situations where GCRT holds and discusses examples where it does not hold. Applications to integral group rings are postponed to a subsequent publication.