ABSTRACT
Under addition ring is an Abelian group, but under multiplication all that’s required is that the operation be closed and associative. It is not required that multiplication be commutative, nor that there be a unity element, and hence nor that multiplicative inverses be present. Rings which satisfy these additional properties are called integral domains. The final step in achieving maximal algebraic structure in a ring is to require not only that ring be an Abelian group under addition, but that its non-zero elements be an Abelian group under multiplication. A field is a commutative ring with a unity element with the additional property that each non-zero element has a multiplicative inverse. The concept of “sub-object” applies to fields just as it does to sets, groups and rings.
