We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the Smith is Huq condition: two surjective left split relations commute as soon as their zero-classes commute.
Martins Ferreira, N., Montoli, A., Ursini, A., & Van der Linden, T. (2017). What is an ideal a zero-class of? Journal of Algebra and its Applications, 16, 1750049. https://doi.org/10.1142/S0219498817500499 (Original work published 2017)