First Isomorphism Theorem

This theorem may be of great use when dealing with computer graphics

The natural homomorphism π is Theorem 6.12 is a special case of more general situation. If f:R → S is a surjective homomorphism of rings, we say that S is homomorphic image of R. if f is a actually an isomorphism (so that S is an isomorphic image of R), then we know that R and S have identical structure. Whenever one of them has a particular algebraic property, the other one has it too. If f is not an isomorphism, then properties of on thing may not hold in the other. However, the properties of S and the homomorphism f often gives us some useful information about R. An analogy with sculpture and photography may be helpful: If f: R → S is an isomorphism, then S is an exact, three-dimensional replica of R. If f is only a surjective homomorphism, then S is a two-dimensional photographic image of F in which some features of R are accurately reflected but others distorted or missing. The next theorem tells us precisely how R, S, and the kernel of f are related in these circumstances.

THEOREM 6.13 (FIRST ISOMORPHISM THEOREM)

Let f:R → S be a surjective homomorphism of rings with kernel K. Then the quotient ring R / K is isomorphic to S.

The theorem states that every homomorphic image of a ring R is isomorphic to a quotient ring R/K for some ideal K. Thus if you know all the quotient rings of R, then you know all the possible homomorphic images of R. The ideal K measures how much information iis lost in passing from the ring R to the homomorphic image R/K. When K = (0_R), then f is an isomorphism by Theorem 6.11, and no information is lost. But when K i large, quite a bit may be lost.