Rings with a bounded number of generators for right ideals

Abstract

Let the ring S S be a finitely generated module over a subring R R of its center. Then it will be shown that S S has the property that every right ideal can be generated by a bounded number of elements if and only if R R has the property that every ideal can be generated by a bounded number of elements. As a corollary we show that a two-sided Noetherian affine ring satisfying a polynomial identity has the property that every right ideal can be generated by a bounded number of elements if and only if every left ideal can be generated by a bounded number of elements.