Quotient_Ring_of_Kernel_of_Ring_Epimorphism

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring epimorphism. Let $K = \map \ker \phi$. Then there is a unique ring isomorphism $g: R_1 / K \to R_2$ such that $g \circ q_K = \phi$ $\phi$ is an isomorphism $K = \set {0_{R_1} }$.