In this article, we use the notions of a semi-open set and topological ideal, in order to define and study a new variant of the classical concept of convergence of sequences in topological spaces, namely, the S-I-convergence. Some basic properties of S-I-convergent sequences and their preservation under certain types of functions are investigated. Also, we study the notions related to compactness and cluster points by using semi-open sets and ideals. Finally, we explore the Iconvergence of sequences in the cartesian product space