In the present work, I have tackled the aforementioned issue from a new perspective, independently of current trends of researches on the emergence of life. The basic idea is as follows. An open reaction network (ORN) into which some chemical substances enter at constant rates can be regarded as a good model of primitive life. It is also probable that primitive life emerged via evolution from an ORN with a simple structure under a prebiotic condition to a complex one involving biopolymers with highly ordered structures and autocatalytic and/or cyclic reactions. Thus, if living things or primitive life demonstrates self-organizing ability leading to a highly ordered and high-function state, we can suppose that an ORN with a simple structure, for example, even one in the initial stage of the above evolution, which may involve neither highly ordered biopolymers nor autocatalytic and/or cyclic reactions, should exhibit such self-organizing ability, because otherwise the above-mentioned evolution could not have occurred. Accordingly, I have aimed to confirm this idea and investigated the fundamental properties of ORNs by paying attention to an ORN with a simple structure. Indeed, such an idea is very unique but I have succeeded in confirming that even an ORN with a simple structure really demonstrates the above-mentioned self-organizing ability when it has reached a nonequilibrium steady state.