Abstract: P. M. Cohn has given a characterization of pseudovaluation on a field in (1). There he defines the notion of an integral element over a multiplicatively closed set and states that the association of the set A to the set A * of all elements that are integral over it is a closure operator. There is an error in the statement that A * = ( A *)*. In this note we show, by an example, that A * and ( A *)* are not equal in general. We also give an example to show that even for the special type of sets that Cohn needs, namely the gauge sets , this property fails to hold. However, since the other conditions for a closure operator ( A ⊆ A * and A ⊆ B implies A * ⊆ B *) are satisfied, one obtains a Kuratowski closure operator by iterating *. If we denote by Ā the closure of A by iterating * then theorem 10·3, its corollary and theorem 10·4 in (1) that have a bearing on this error, are valid as stated with the same proof replacing only K * by . Further theorem 13·3 is valid without modification since Cohn considers only a special class of pseudovaluations namely regular pseudo valuations for which K * = .
Publication Year: 1975
Publication Date: 1975-09-01
Language: en
Type: article
Indexed In: ['crossref']
Access and Citation
Cited By Count: 1
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot