Let \(A\) be a Noetherian local ring with maximal ideal \(\mathfrak m\), \(I\) a proper ideal and \(M\) a finitely generated \(A/I\)-module. Then the following inequality holds: \[ \beta_p^{A/I}(M) \leq \begin{cases} \beta_p^A(M) & (p = 0, 1) \\ \beta_p^A(M) + \sum_{i = 0}^{p-2} \beta_{p-i-2}^A(I) \beta_i^{A/I}(M) & (p \geq 2). \end{cases} \tag{1} \] Here \(\beta_p^A(M)\) denotes the \(p\)-th Betti number of \(M\) over \(A\). We say that \((M, I)\) is a Golod pair if equality in (1) holds. In this case the author constructed the minimal free resolution of \(M\) over \(A/I\) by using ones of \(M\) and \(I\) over \(A\) [D. Gokhale, Commun. Algebra 22, No. 3, 989-1030 (1994; Zbl 0796.13010)]. We know that \((A/\mathfrak m, I)\) is a Golod pair if \(I\) is the product of a proper ideal and an \(A\)-non zero divisor [see L. L. Avramov, J. Algebra 50, 400-453 (1978; Zbl 0395.13005); proposition 4.4]. In the present paper the author improves this. Let \(J\) be a proper ideal and \(x\) an \(A\)-non zero divisor. Then \((M, xJ)\) is a Golod pair for any finitely generated \(A/xJ\)-module \(M\).