# 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?

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

1. $$++$$ is associative.
2. There exists $$0∈R0\in R$$ such that $$0+a=a+0=a0+a=a+0=a$$ for all $$a∈Ra\in R$$.
3. For every $$a∈Ra\in R$$ there exists $$b∈Rb\in R$$ such that $$a+b=b+a=0a+b=b+a=0$$.
4. $$++$$ is commutative.
5. $$×\times$$ is associative.
6. $$×\times$$ distributes over $$++$$ on the left: for all $$a,b,c∈Ra,b,c\in R$$, $$a×(b+c)=(a×b)+(a×c)a\times(b+c) = (a\times b)+(a\times c)$$.
7. $$×\times$$ distributes over $$++$$ on the right: for all $$a,b,c∈Ra,b,c\in R$$, $$(b+c)×a=(b×a)+(c×a)(b+c)\times a = (b\times a)+(c\times a)$$.

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

We also have the following items:

a. There exists $$1∈R1\in R$$ such that $$1×a=a×1=a1\times a = a\times 1 = a$$ for all $$a∈Ra\in R$$.

b. $$1≠01\neq 0$$.

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

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 $$(R−{0},×)(R-\{0\},\times)$$ is a group (note that we need to remove $$00$$ 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 $$(R,+)(R,+)$$ is an abelian group, that $$(R−{0},×)(R-\{0\},\times)$$ is an abelian group, and that these structures “mesh together” via (6) and (7). In a ring, we have that $$(R,+)(R,+)$$ is an abelian group, that $$(R,×)(R,\times)$$ is a semigroup (or better yet, a semigroup with $$00$$), 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 $$n×nn\times n$$ matrices with coefficients in, say, $$R\mathbb{R}$$, $$n>1n\gt 1$$, if you want noncommutativity); every ring with unity is a ring, but there are rings that are not rings with unity (strictly upper triangular $$3×33\times 3$$ matrices with coefficients in $$R\mathbb{R}$$, for instance). So
$$Fields⊊Division rings⊊Rings with unity⊊Rings\text{Fields}\subsetneq \text{Division rings}\subsetneq \text{Rings with unity} \subsetneq \text{Rings}$$
and
$$Fields⊊Commutative rings with unity⊊Commutative rings⊊Rings.\text{Fields}\subsetneq \text{Commutative rings with unity}\subsetneq \text{Commutative rings}\subsetneq \text{Rings}.$$