We introduce a special kind of fusion of a link and a trivial link, called a simple ribbon fusion, and consider how it affects on the genus of a link. As for a genus of a link, C.Goldberg introduced two notions, called the disconnectivity number and the r-th genus of a link in his PhD thesis of 1970: the disconnectivity number v(L) of a link L is the maximal number of connected components of all the Seifert surfaces for L, and the r-th genus of L is the minimal number of genera of all the Seifert surfaces for L with r connected components, where r is a natural number which is less than or equal to v(L). We show that the disconnectivity number never increases and the r-th genus is never decreases by a simple ribbon fusion. We also give a necessary and sufficient condition for the equalities to hold.