A consequence of using non-unital rings is that you have to postulate that its additive structure is commutative, so that the group additive structure is abelian. This is not necessary for unital rings, since the commutativity of addition is a consequence of the associativity of addition, of the definition of the multiplicative neutral element and of the left and rigth distributivity of addition for multiplication. I do not know if this fact can be considered as a reason for considering rings without identity. Gérard Lang

A consequence of using non-unital rings is that you have to postulate that its additive structure is commutative, so that the group additive structure is abelian. This is not necessary for unital rings, since the commutativity of addition is a consequence of the associativity of addition, of the definition of the multiplicative neutral element and of the left and right distributivity of addition for multiplication. I do not know if this fact can be considered as a reason for considering rings without identity. Gérard Lang