Left serial rings and their factor skew fields
A (possibly skew) field \(D\) is said to be of type 1 if it has no proper subfield \(D'\) isomorphic to \(D\) such that the left degree \([D:D']\), is finite, and of type 2 otherwise. Examples of both are given and there is a brief discussion of the (still open) question whether taking finite extensions preserves the type. Given a (skew) field \(D\), a subfield \(D'\) is said to be inner in \(D\) if \(D\) is spanned over \(D'\) by the normalizer of \(D'\), and any \(D'\)-basis in the normalizer is called an inner basis. Inner bases for subfields of the field of fractions of a skew polynomial ring are briefly considered, as a preparation to study uniserial rings. For a local ring \(R\) with residue class field of type 1 the following are shown to be equivalent (a) \(R\) is uniserial (b) either its left or right (or both) indecomposable injective modules are uniserial. These results are used to prove that for an Artinian ring \(R\) which modulo its Jacobson radical is finitely generated over its centre, \(R\) is right serial if and only if every indecomposable injective left \(R\)-module is uniserial. Here the condition of finite generation cannot be omitted, by examples of W. Xue (to appear).
scientific article; zbMATH DE number 35997
scientific article