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  2. A RING is a set equipped with two operations, called addition and multiplication. A RING is a GROUP under addition and satisfies some of the properties of a group for multiplication. A FIELD is a GROUP under both addition and multiplication.
    www-users.cse.umn.edu/~brubaker/docs/152/152g…
    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.
    math.stackexchange.com/questions/141249/what-i…
    In fact, every ring is a group, and every field is a ring. A ring is an abelian group with an additional operation, where the second operation is associative and the distributive property make the two operations "compatible".
    math.stackexchange.com/questions/75/what-are-th…

    A ring is a group under addition. A field is a group under addition and a group under multiplication. Any further description tends to be more confusing. One big difference is that a ring need not be commutative under multiplication, whereas a field is.

    www.physicsforums.com/threads/whats-the-differe…
    A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".
    en.wikipedia.org/wiki/Division_ring
     
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