On the cardinality of ideals in Artinian rings
Let \(R\) be a right Artinian ring and let \(\alpha\) be an infinite cardinal number. The authors' main result shows that \(R\) contains two uniquely determined (two-sided) ideals \(S(\alpha)\) and \(S^*(\alpha)\) such that: (i) \(S(\alpha)\) is the largest right ideal of \(R\) with cardinality less than \(\alpha\), (ii) \(S^*(\alpha)\) is the smallest ideal of \(R\) such that \(R/S^*(\alpha)\) has cardinality less than \(\alpha\). Moreover, \(S^*(\alpha)\) has a right identity, (so that it is also right Artinian) and \(S^*(\alpha)\cap S(\alpha)=0\). Also, letting \(T(\alpha)\) be the sum of all the left ideals of \(R\) with cardinality less than \(\alpha\), if \(\text{card}\,T(\alpha)<\alpha\) (as is the case if \(R\) is also left Artinian) then \(T(\alpha)=S(\alpha)\) and \(R\) is the direct sum of \(S(\alpha)\) and \(S^*(\alpha)\). The authors show by example that this cardinality condition on \(T(\alpha)\) can not be omitted. As a consequence of these results the authors obtain several results on the structure of right Artinian rings. For example, they show that if \(R\) is right Artinian such that no two ideals of \(R\) have the same cardinality then \(R\) is the direct sum of a semiprime Artinian ring and a finite ring.
scientific article; zbMATH DE number 4075312