ABSTRACT
19.1 For groups we have seen that normal subgroups are necessary in order to introduce a natural group structure on the quotient set with respect to the corresponding congruence relation. Since rings have an abelian additive structure, there is not any problem concerning “normality” in order to have an additive structure on suitable quotient structures. The problem obviously resides in the multiplicative structure which is not necessarily commutative. Just like a normal subgroup made a difference between right and left cosets disappear, the two-sidedness of a two-sided ideal allows the definition of a ring structure on the quotient group already defined with respect to it. So, in a sense, one could think of two-sided ideals as replacing the notion of “normal subring”. The quotient ring has a universal property (Theorem 19.5) as in the group case and again the three isomorphism theorems follow from this (19.6, 19.8 and 19.11). The canonical map from ℤ to any ring R, defined by sending 1ℤ to 1 R , gives rise to the definition of prime subring and characteristic of the ring {19.15). The use of rings of positive characteristic also allows us to proof Fermat's theorem (Theorem 19.23), also known as “The little Fermat's Theorem”.
