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?

Answer

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

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

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

We also have the following items:

a. There exists 1R such that 1×a=a×1=a for all aR.

b. 10.

c. For every aR, a0, there exists bR such that a×b=b×a=1.

d. × 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},×) 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},×) 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,×) 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×n matrices with coefficients in, say, R, n>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 3×3 matrices with coefficients in R, for instance). So
FieldsDivision ringsRings with unityRings
and
FieldsCommutative rings with unityCommutative ringsRings.

Attribution
Source : Link , Question Author : zinking , Answer Author : Arturo Magidin

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