Abstract Algebra: Definition of Subrings and Examples of Ideals in Ring Theory

Subrings are subsets of rings, R, that are themselves rings with respect to the addition and multiplication that are defined on R. This video shows that a subset A of a ring R is a subring of R if it is closed with respect to the addition and multiplication that are defined on R, and if A is closed with respect to additive inverses. It is noted that since the ring R is abelian with respect to addition, that every subring of R is a normal subgroup of R with respect to addition. Therefore, every subring of a ring gives rise to a quotient group of R with respect to addition. The question is raised as to which of these subrings provides enough structure to permit the cosets that comprise the elements of this quotient group to form a quotient ring in a natural way. An answer is provided to this question: if the product of elements of A with elements of R produce elements of A, then the cosets of A form a ring with respect to the natural operations of coset addition and multiplication on R/A.