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

CogitoErgoCogitoSum
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cobbal
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    Is there a diagram somewhere that depicts the relationships pictorially? – occulus Aug 29 '14 at 09:11
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    Ah, this page contains some useful diagrams concerning group etc. relationships: http://en.wikipedia.org/wiki/Magma_(algebra) – occulus Aug 29 '14 at 09:30

6 Answers6

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They should feel similar! 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".

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

Joe
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BBischof
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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.

Ittay Weiss
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François G. Dorais
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    A ring does not necessarily have a multiplicative identity. – Andrew Maurer Jan 07 '13 at 01:59
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    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 07 '13 at 14:34
  • "if you forgot about addition, then the ring does not become a group" ... and since it is no longer a ring (having only one operation), it becomes a "Magma"? – Les Aug 04 '18 at 15:25
  • @Les It is a magma since it has one binary operation, but moreover it is a monoid since it is associative and 1 acts as an identity element. In the case of a field, it's actually one shy of a group in the sense that 0 is the only element without an inverse. – François G. Dorais Aug 06 '18 at 22:13
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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.)

MJD
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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.

Akhil Mathew
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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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Let $G$ be a set. A binary relation on $G$ is a function $\star$ from $G\times G$ to $G$. Typically, we use infix notation to denote this function rather than prefix notation; that is, we write $a\star b$ instead of $\star(a,b)$.

A group is an ordered pair $(G,\star)$, where the binary relation $\star:G\times G\to G$ satisfies the following "group axioms":

  1. Associativity: for all $a,b,c\in G$, we have $(a\star b)\star c=a\star(b\star c)$.
  2. Existence of an identity element: there is an $b\in G$ such that $a\star b=b\star a=a$ for all $a\in G$. It can be proven that this element is unique. It is called the identity element (or neutral element) of the group, and is denoted as $e$.
  3. Existence of inverse elements: for all $x\in G$, there is a $y\in G$ such that $x\star y=y\star x=e$. It can be proven that for each $x$, this element $y$ is unique. It is called the inverse of $x$, and is denoted as $x^{-1}$.

Examples of groups include $(\mathbb Z,+)$ and $(\mathbb Q-\{0\},\cdot)$, where $+$ and $\cdot$ refer to the usual addition and multiplication operations, respectively. It is worth noting that in $(\mathbb Z,+)$, we follow a slightly different notational convention: the identity is written as $0$, and the inverse of $x$ is written as $-x$. This is known as additive notation, and it is common to use this whenever the group operation is written as $+$. In $(\mathbb Q-\{0\},\cdot)$, we use multiplicative notation, where the identity is written as $1$ (but the notation for inverses is the same). Again, it is standard to use multiplicative notation whenenever the group operation is written as $\cdot$ or $\times$.

Finally, while strictly speaking a group is an ordered pair $(G,\star)$, it is very common to call the set $G$ a group. If it is unclear from context what the binary operation is, we say that $G$ is a group under $\star$. For instance, we would say that $\mathbb Z$ is a group under $+$.

An abelian group is a group with a commutative binary operation: for all $a,b\in G$, we have $a\star b = b\star a$.

A field is an ordered triple $(F,+,\cdot)$, where $+$ and $\cdot$ are binary relations on $F$ satisfying the following "field axioms":

  1. The ordered pair $(F,+)$ is an abelian group.
  2. The ordered pair $(F-\{0\},\cdot)$ is an abelian group (here, $0$ denotes the identity element of $(F,+)$).
  3. The operation $\cdot$ is distributive over $+$: for all $a,b,c\in F$, we have $a\cdot(b+c)=(b+c)\cdot a=(a\cdot b)+(a\cdot c)$.

The operation $+$ is called addition and $\cdot$ is called multiplication.

A ring is an ordered triple $(R,+,\cdot)$. Rings have a similar definition to fields, except that we impose fewer requirements on the ordered pair $(F-\{0\},\cdot)$: it now only has to be an associative structure, rather than being an abelian group. Aside from this, rings and fields have the same definition. A ring with an identity element is called a ring with identity.

Like with many mathematical definitions, there are slight variations in usage. Some authors require that all rings have an identity element, and describe what we called a ring a rng (the omission of the letter i signals that it is not necessary for there to be an identity element).

Joe
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