What's the difference between a ring and a field?

[swampwiz]

They seem to mean the almost the same thing, with a field allowing subtraction & division whereas a ring only needs addition & multiplication. Is that a ring can mean "addition" & "multiplication" in some way that isn't between numbers per se, but between whatever abstract things can be dreamed up, whereas a field is strictly for real/complex numbers?

 

[Mark44]

They seem to mean the almost the same thing, with a field allowing subtraction & division whereas a ring only needs addition & multiplication.

No, they are different. Here are partial definitions from "CRC Standard Mathematical Tables."
Ring - a system $[R, \theta, A]$, where R consists of a nonempty set, $\theta$ consists of two binary operations, + and $\times$, and A is a set of axioms.

Example: the set of integers, with the usual operations of addition and multiplication, is a ring.
I'll let you look up the axioms, which you should be able to find on Wikipedia or elsewhere.

Integral Domain - a ring R in which multiplication satisfies additional assumptions beyond those of a ring, such as multiplication being commutative.
Example: As it turns out, the ring of integers also satisfies the additional axioms, and is an integral domain.

Field - an integral domain in which every element except $z$ is a unit. Here, $z$ is the element that plays the role of zero.
Examples: the field of rational numbers, with the usual operations of addition and multiplication.
The field of real numbers, with the usual operations of addition and multiplication.

BTW, the operations are addition and multiplication. Subtraction such as a - b is defined in terms of adding the additive inverse. I.e., a - b is defined as a + (-b), where -b is the thing that can be added to b to produce the additive identity.

Is that a ring can mean "addition" & "multiplication" in some way that isn't between numbers per se, but between whatever abstract things can be dreamed up, whereas a field is strictly for real/complex numbers?

No, as mentioned above, there is the field of rational numbers. There are also finite fields, also called Galois fields.

 

[fresh_42]

They seem to mean the almost the same thing, with a field allowing subtraction & division whereas a ring only needs addition & multiplication. Is that a ring can mean "addition" & "multiplication" in some way that isn't between numbers per se, but between whatever abstract things can be dreamed up, whereas a field is strictly for real/complex numbers?

Neither nor. You are right that fields allow divisions whereas rings in general do not, will say a field is always a ring but not the other way around.

Anyway, let's make an example:

• The set $\mathbb{Z}_4 = \{\,0,1,2,3\,\}$ of possible remainders of a division by four builds a commutative ring. But we have $2\cdot 2=0$ in this ring, so we cannot divide by $2$.
• The set $\mathbb{Z}_5=\{\,0,1,2,3,4\,\}$ of possible remainders of a division by five builds a commutative ring which is also a field. All elements have multiplicative inverses.

Polynomials build rings which aren't fields. Other examples are matrix rings. E.g.
$$
\left\{\,\left. \begin{bmatrix}0&m&n\\0&0&k\\0&0&0\end{bmatrix} \,\right|\,k,m,n\in \mathbb{Z} \,\right\}
$$
builds a ring, one that has no multiplicative neutral element, a $1$.

Rings in which there are no elements $a\cdot b=0$ other than $0$ can always be extended to a field, like the integers are extended to the rationals. But this does not mean the elements are what you thought of, this is also true for polynomials with coefficients in say $\mathbb{Z}_5$ or $\{\,0,1\,\}$ if $1+1=0$.

Neither ring nor field are bounded to what you call numbers. Those are only examples, not representatives.

 

[swampwiz]

OK, other than the fact that a field is always a ring (but not vice-versa), I am hopelessly confused.

 

[fresh_42]

Consider the examples $\mathbb{Z}_4$ and $\mathbb{Z}_5$ I gave.

In a division by, say four, we have the remainders $0,1,2,3$. Now try to construct the addition and multiplication tables. At the end, look up whether $1$ occurred in the row of $2$ in the multiplication table. If not, then there cannot be an inverse. Then do the same with $0,1,2,3,4$ as remainders of the division by $5$ and check, whether there is a $1$ in every row of the multiplication table, except the row of $0$ of course, which strictly speaking doesn't belong in the multiplication table.

 

[mathman]

In a ring you have addition, subtraction, and multiplication. A field has division as well. A ring is a group under addition. A field is a group under addition and a group under multiplication. Any further description tends to be more confusing.

 

[phyzguy]

One big difference is that a ring need not be commutative under multiplication, whereas a field is.

 

[WWGD]

In a field, every nonzero
element has an inverse ,both additive and multiplicative, not so in a ring. In a field, if ab=0 then either a=0 or b=0, not so in a ring, e.g. in $\mathbb Z_6 ,[2][3]=[0]$.Notice this last is not possible in $\mathbb Z_p$ , p being a prime makes this last impossible.

 

[swampwiz]

In a field, every nonzero
element has an inverse ,both additive and multiplicative, not so in a ring. In a field, if ab=0 then either a=0 or b=0, not so in a ring, e.g. in $\mathbb Z_6 ,[2][3]=[0]$.Notice this last is not possible in $\mathbb Z_p$ , p being a prime makes this last impossible.

Is it that a field acts like a regular number?

 

[WWGD]

To

Is it that a field acts like a regular number?

In the sense that the product of nonzero numbers cannot be zero, yes.