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?

There's a whole range of algebraic structures. Perhaps the 5 best known are semigroups, monoids, groups, rings, and fields.


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*A semigroup is a set with a closed, associative, binary operation.

*A monoid is a semigroup with an identity element.

*A group is a monoid with inverse elements.

*An abelian group is a group where the binary operation is commutative.

*A ring is an abelian group (under addition, say) that happens to have a second closed, associative, binary operation as well. And these two operations satisfy a distribution law. (You may or may not require rings to have an identity with the second operation)

*A field is a ring where both operations commute, where every element has both an additive (i.e. the first operation) and a multiplicative (i.e. the second operation) inverse (and thus there is a multiplicative identity), and the extra requirement that if $xy = 0$ for some $x \not = 0$, then we must have $y = 0$ (we call this having no zero-divisors).


People study these, and maps between them, because it is stunning how often things can be given a group or ring-like structure. So knowing how these things behave carries a lot of information about many things.
