Display omitted •We made a new approach to the classical ring theory via soft set theory, with the concept of soft intersection rings.•We defined ideals, (generalized) bi-ideals, interior ideals and quasi-ideals of soft rings.•By defining soft union–intersection product, we obtain the relationship between this new concept and soft intersection ring and its different ideals.•We also characterized regular, regular duo, intra-regular and strongly regular rings by soft intersection rings and ideals. In this paper, we make a completely new approach to the classical ring theory via soft set theory, with the concept of soft intersection rings, ideals, (generalized) bi-ideals, interior ideals and quasi-ideals. Particularly, we define soft union–intersection product and obtain the relationship between this new concept and soft intersection ring and its different ideals. Moreover, we characterize regular, regular duo, intra-regular and strongly regular rings by soft intersection rings and ideals.