On the \(GE_ 2\) of graded rings
A ring R is said to be a G\(E_ 2\)-ring if \(GL_ 2(R)\), the group of 2\(\times 2\) invertible matrices over R, is generated by elementary and diagonal matrices. The author obtains some necessary conditions for a graded domain to be a \(GE_ 2\)-ring. This gives rise to a number of interesting consequences. Amongst other things it is proved, for example, that the polynomial ring D[X] over a commutative Noetherian ring D is a \(GE_ 2\)-ring iff every prime ideal in D is maximal (i.e. D is Artinian). Another result of interest is that the Laurent polynomial ring \(D[X,X^{-1}]\) over a commutative valuation domain D is a \(GE_ 2\)- ring. (Reviewer's remark: As far as I know it is not known if the Laurent polynomial ring over the rational integers is a \(GE_ 2\)-ring.)
scientific article; zbMATH DE number 3859404