The author proves two theorems. The first states that if L is a left ideal of the nth Weyl algebra \(A_ n\) over a field k of characteristic 0 then \(k+L\) is a finitely generated subalgebra. This is a special case of an unpublished result of L. W. Small who proved it for any simple left noetherian k-algebra by using ideas from his paper with S. Montgomery [Bull. Lond. Math. Soc. 13, 33-38 (1981; Zbl 0453.16021)]. The second theorem states that if c is a regular element of positive degree in a graded k-algebra A such that \(k+Ac\) is left noetherian then its centralizer \(C_ A(c)\) is a finitely generated left k[c]-module.