Fundamental Theorem of Ring Homomorphisms: Identifying the homomorphic images of a ring

If h is a surjective ring homomorphism from a ring R1 to a ring r2, then there is some set of elements of the ring R1 that gets sent to the zero element (the additive identity) of the ring R2. This set of domain elements is traditionally called the kernel of the homomorphism h. We call this set the vanishing set of h since being sent to zero is akin to vanishing. We saw in a previous video that the vanishing set of h is a subring of the domain R1. Moreover, if a is any member of the vanishing set of h and x is any member of R1, then a*x and x*a are both members of the vanishing set of h. In other words, the vanishing set K of h has the property that it absorbs products. That is, the vanishing set of h is an ideal of R1. Therefore, R1/K is a ring. We discuss in this video that R1/K is isomorphic to R2. In other words, every homomorphic image of R1 is isomorphic to a quotient ring of R1. Moreover, every quotient ring of R1 is a homomorphic image of R1. Therefore, there is a one-to-one correspondence between the homomorphic images of a ring, and the quotient rings of that ring. This result is known as the Fundamental Theorem of Ring Homomorphisms. It stands as one of the hallmarks of the study of Ring Theory.