What are the differences between rings, groups, and fields?

Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?

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Is there a diagram somewhere that depicts the relationships pictorially? –  occulus Aug 29 at 9:11
Ah, this page contains some useful diagrams concerning group etc. relationships: en.wikipedia.org/wiki/Magma_(algebra) –  occulus Aug 29 at 9:30

They should feel similar! In fact, every ring is a group, and every field is a ring. A ring is a group with an additional operation, where the second operation is associative and the distributive properties make the two operations "compatible".

A field is a ring such that the second operation also satisfies all the group properties (after throwing out the additive identity); i.e. it has multiplicative inverses, multiplicative identity, and is commutative.

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Please note that the multiplicative group of a field (obviously) does not contain the zero from the field in question as there is no inverse. –  Pieter Jul 20 '10 at 20:04
Yes, this is a point that must be made! –  BBischof Jul 20 '10 at 20:10
The phrase "i.e.,... is commutative" is somewhat confusing, as it suggests that being commutative is a group property. Commutativity is actually an additional property to the one requiring nonzero elements to form a multiplicative group; without it one has a division ring. –  Marc van Leeuwen Feb 10 '13 at 17:59
It should be specified that addition in a ring is commutative. –  Andreas Caranti Feb 10 '13 at 18:01

I won't explain what a ring or a group is, because that's already been done, but I'll add something else. One reason groups and rings feel similar is that they are both "algebraic structures" in the sense of universal algebra. So for instance, the operation of quotienting via a normal subgroup (for a group) and a two-sided ideal (for a ring) are basically instances of quotienting via an invariant equivalence relation in universal algebra. A field, by contrast, is not really a construction of universal algebra (because the operation $x \to x^{-1}$ is not everywhere defined) -- which is why free fields don't exist, for instance -- though they are a special case of rings.

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You're right to think that the definitions are very similar. The main difference between groups and rings is that rings have two binary operations (usually called addition and multiplication) instead of just one binary operation.

If you forget about multiplication, then a ring becomes a group with respect to addition (the identity is 0 and inverses are negatives). This group is always commutative!

If you forget about addition, then a ring does not become a group with respect to multiplication. The binary operation of multiplication is associative and it does have an identity 1, but some elements like 0 do not have inverses. (This structure is called a monoid.)

A commutative ring is a field when all nonzero elements have multiplicative inverses. In this case, if you forget about addition and remove 0, the remaining elements do form a group under multiplication. This group is again commutative.

A division ring is a (not necessarily commutative) ring in which all nonzero elements have multiplicative inverses. Again, if you forget about addition and remove 0, the remaining elements do form a group under multiplication. This group is not necessarily commutative. An example of a division ring which is not a field are the quaternions.

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A ring does not necessarily have a multiplicative identity. –  andybenji Jan 7 '13 at 1:59
That used to be the case but most authors today define a ring to have $1$. The unusual looking term rng is sometimes used for the concept without $1$. –  François G. Dorais Jan 7 '13 at 14:34

Any group $G$ is isomorphic to its opposite group $G^{\text{op}}$ via the map $g \mapsto g^{-1}$, however there is no such natural map for rings and in general it is not true that a ring is isomorphic to its opposite ring.

Therefore, it is always possible to obtain a right action of a group $G$ if a left action is given whereas it may not be possible to equip a left $R$-module with a right $R$-module structure.

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A group is an abstraction of addition and subtraction—except that the group operation might not be commutative. But the important part is that there is an operation, which is something like addition, and the operation can be reversed, so there is also something like subtraction.

To this, a ring adds multiplication, but not necessarily division.

To this, a field adds division.

(Going the other way from a group, we have the monoid, which has addition, but not subtraction.)

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