A good theory of ideals in regular multi-pointed categories

We extend the notion of an ‘‘ideal’’ from regular pointed categories to regular multi-pointed categories, and having ‘‘a good theory of ideals’’ will mean that there is a bijection between ideals and kernel pairs, which in the pointed case is the main property of ideal determined categories. The study of general categories with a good theory of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat categories:weprove that in the case when all morphisms are chosen as null morphisms, the presence of a good theory of ideals becomes precisely the property for a regular category to be a Barr exact Goursat category.