Commutative rings whose proper ideals are serial
Let \(R\) be a commutative ring with identity and \(M\) be an \(R\)-module. Then \(M\) is called uniserial, if its submodules form a chain under inclusion. Also \(M\) is called serial, when it is a direct sum of uniserial modules. If \(R\) is a uniserial (resp., serial) \(R\)-module, then it is said that \(R\) is a uniserial (resp., serial) ring. In this paper, commutative rings in which every proper ideal is serial are characterized. The authors show that if \((R,\mathfrak{M})\) is a local ring, then every proper ideal of \(R\) is serial, if and only if \(\mathfrak{M}\) is a direct sum of a uniserial module and a semisimple module. They also present examples of local rings in which every proper ideal is serial, but the ring itself is not serial. If \(R\) is not a local ring, the authors prove that every proper ideal of \(R\) is serial, if and only if \(R\) is a serial ring. Moreover, they show that in this case there exists an integer \(n\) such that every ideal of \(R\) is a direct sum of at most \(n\) uniserial \(R\)-module.
scientific article; zbMATH DE number 6839441
scientific article