What is difference between a ring and a field?

Question

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?

Answer 1

A ring is an ordered triple, $(R,+,\times)$, where $R$ is a set, $+\colon R\times R\to R$ and $\times\colon R\times R\to R$ are binary operations (usually written in in-fix notation) such that:

1. $+$ is associative.
2. There exists $0\in R$ such that $0+a=a+0=a$ for all $a\in R$.
3. For every $a\in R$ there exists $b\in R$ such that $a+b=b+a=0$.
4. $+$ is commutative.
5. $\times$ is associative.
6. $\times$ distributes over $+$ on the left: for all $a,b,c\in R$, $a\times(b+c) = (a\times b)+(a\times c)$.
7. $\times$ distributes over $+$ on the right: for all $a,b,c\in R$, $(b+c)\times a = (b\times a)+(c\times a)$.

1-4 tell us that $(R,+)$ is an abelian group. 5 tells us that $(R,\times)$ is a semigroup. 6 and 7 are the two distributive laws that you mention.

We also have the following items:

a. There exists $1\in R$ such that $1\times a = a\times 1 = a$ for all $a\in R$.

b. $1\neq 0$.

c. For every $a\in R$, $a\neq 0$, there exists $b\in R$ such that $a\times b = b\times a = 1$.

d. $\times$ is commutative.

A ring that satisfies (1)-(7)+(a) is said to be a "ring with unity." Clearly, every ring with unity is also a ring; it takes "more" to be a ring with unity than to be a ring.

A ring that satisfies (1)-(7)+(a,b,c) is said to be a division ring. Again, eveyr division ring is a ring, and it takes "more" to be a division ring than to be a ring. (5)+(a)+(b)+(c) tell us that $(R-\{0\},\times)$ is a group (note that we need to remove $0$ because (c) specifies nonzero, and we need (b) to ensure we are left with something).

A ring that satisfies (1)-(7)+(a,b,c,d) is a field. Again, every field is a ring.

We do indeed have that $(R,+)$ is an abelian group, that $(R-\{0\},\times)$ is an abelian group, and that these structures "mesh together" via (6) and (7). In a ring, we have that $(R,+)$ is an abelian group, that $(R,\times)$ is a semigroup (or better yet, a semigroup with $0$), and that the two structures "mesh well".

We have that every field is a division ring, but there are division rings that are not fields (e.g., the quaternions); every division ring is a ring with unity, but there are rings with unity that are not division rings (e.g., the integers if you want commutativity, the $n\times n$ matrices with coefficients in, say, $\mathbb{R}$, $n\gt 1$, if you want noncommutativity); every ring with unity is a ring, but there are rings that are not rings with unity (strictly upper triangular $3\times 3$ matrices with coefficients in $\mathbb{R}$, for instance). So $$\text{Fields}\subsetneq \text{Division rings}\subsetneq \text{Rings with unity} \subsetneq \text{Rings}$$ and $$\text{Fields}\subsetneq \text{Commutative rings with unity}\subsetneq \text{Commutative rings}\subsetneq \text{Rings}.$$

Answer 2

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

• 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.

Answer 3

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