Abstract Algebra | 12. Subrings

In this video we define and discuss the notion of subring of a ring and give examples of subrings. We also prove theorems which allow us to more easily check whether a subset of a ring is a subring. Important note: We are assuming our rings do not necessarily have multiplicative identity. In texts where existence of a multiplicative identity is included in the definition of a ring, the multiplicative identity must also be present in any subring. For example, using our definitions, the set of even integers is a subring of the set of integers. In the setting where we assume that rings always must contain a multiplicative identity, we would not view the even integers as a subring of the integers since 1 is not in the set of even integers.