What is difference between a ring and a field? The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. 
A field can be thought of as two groups with extra distributivity law. 
A ring is more complex: with abelian group and a semigroup with extra distributivity law.
Is a ring a more basic structure than a field, or vice versa? What's the relation between them? What's the background why people study them?

A field has multiplicative inverses, rings don't need to have that- Just additive ones. Rings are the more basic object. ${Fields}\subset {Rings}$
