A ring is a set equipped with two operations (usually referred to as addition and multiplication) that satisfy certain properties: there are additive and multiplicative identities and additive inverses, addition is commutative, and the operations are associative and distributive.
The study of rings has its roots in algebraic number theory, via rings that are generalizations and extensions of the integers, as well as algebraic geometry, via rings of polynomials. These kinds of rings can be used to solve a variety of problems in number theory and algebra; one of the earliest such applications was the use of the Gaussian integers by Fermat, to prove his famous two-square theorem. There are many examples of rings in other areas of mathematics as well, including topology and mathematical analysis.
A ring is a set together with two operations and satisfying the following properties (ring axioms):
(1) is an abelian group under addition. That is, is closed under addition, there is an additive identity (called ), every element has an additive inverse , and addition is associative and commutative.
(2) is closed under multiplication, and multiplication is associative:
(3) Multiplication distributes over addition:
A ring is usually denoted by and often it is written only as when the operations are understood.
(1) There are two further requirements one might impose on a ring that lead to interesting classes of rings. For instance, if multiplication is commutative, the ring is called a commutative ring. The theory of commutative rings differs quite significantly from the the theory of non-commutative rings; commutative rings are better understood and have been more extensively studied. Most of the examples and results in this wiki will be for commutative rings. Again there may be an element in such that for all elements in , . If such an element exists, we call it the unity of the ring, and the ring is called a ring with unity. Else it is called a ring without unity or a "rng" (a ring without ).
(2) If is a commutative ring and such that and , then and are said to be divisors of . If in a commutative ring with unity, there is no divisor of the additive identity, i.e. , then is said to be an integral domain. Thus a commutative ring with unity is said to be an integral domain if for all elements in , implies either or .
(3) If every nonzero element in a commutative ring with unity has a multiplicative inverse as well, the ring is called a field. Fields are fundamental objects in number theory, algebraic geometry, and many other areas of mathematics. If every nonzero element in a ring with unity has a multiplicative inverse, the ring is called a division ring or a skew field. A field is thus a commutative skew field. Non-commutative ones are called strictly skew fields.
This section lists many of the common rings and classes of rings that arise in various mathematical contexts.
(1) The ring of integers is the canonical example of a ring. It is an easy exercise to see that is an integral domain but not a field.
(2) There are many other similar rings studied in algebraic number theory, of the form , where is an algebraic integer. For example, is a ring, an integral domain, to be precise. Also we have the ring of Gaussian integers , where is the imaginary unit.
(3) If is a ring, then so is the ring of polynomials with coefficients in . In particular, when is the finite field with elements, has many similarities with . For example, there is a Euclidean algorithm and hence unique factorization into irreducibles. See the introduction to algebraic number theory for details.
More generally, if is a set and is a ring, the set of functions from to is a ring, with the natural operations of pointwise addition and multiplication of functions. For many sets , this ring has many interesting subrings constructed by restricting to functions with properties that are preserved under addition and multiplication. If , for instance, there are subrings of continuous functions, differentiable functions, polynomial functions, and so on.
(4) The set of matrices with entries in a commutative ring is a ring, which is non-commutative for . This ring has a unity, the identity matrix. But it may have divisors of zero. E.g. . This shows that and are divisors of zero in the ring .
(5) Another classical example is the ring of quaternions, the set of expressions of the form , where and satisfy the relations This has numerous applications in physics. This is a strictly skew field.
In this section, for simplicity's sake, all rings will be assumed to be commutative. (There are generalizations of these ideas to non-commutative rings, but the definitions are more unwieldy.)
An ideal in a commutative ring is a nonempty set that
(1) is closed under addition
(2) "swallows up" under multiplication: if and , then .
If , the set is an ideal, and is called the ideal generated by the .
The ideal generated by one element, , the set of multiples of , is called a principal ideal. A ring in which every ideal is principal is called a principal ideal ring.
Show that is a principal ideal ring.
If is a field, show that is a principal ideal ring.
Show that is not a principal ideal ring.
Partial solution: Any ring with a reasonable division algorithm is called a Euclidean ring (after the Euclidean algorithm). All such rings are principal ideal rings; the idea is to take the "smallest" nonzero element in the ideal and then use the division algorithm to show that every other element in the ideal is a multiple of . This is because if where is smaller than , then is in the ideal but is smaller than , so it must be . Since and both have division algorithms, the first two results follow.
In , the ideal is not principal. (Exercise for the reader. Hint: consider the degree of a generator.)
Given a ring and an ideal , there is an object called the quotient ring . The example to keep in mind is and the ideal generated by an integer . Then is the familiar ring of integers mod .
The ring is the set of elements , where . Two expressions and are equal in if and only if . Elements are added and multiplied just as they are in : and .
The subtle part of this definition is that it is well-defined: that is, the arithmetic in gives the same results no matter which representative of an element is picked. (Again, the example to keep in mind is .) The proof of this well-definition uses the properties of the ideal in an essential way (and is left as an exercise for the reader).
If and , then as remarked earlier.
If and , then can be identified with , by identifying with . (Note that in , so is a square root of .)
If and , then can be identified with Gaussian Integers, , by identifying with .
If in and and are nonzero, then and are called zero-divisors. A ring with no zero-divisors is called a domain, and a commutative domain is called an integral domain.
For which is an integral domain?
If is composite (where ), then mod but and are nonzero mod because they are strictly smaller than . So is not an integral domain when is composite.
On the other hand, if is prime and mod , then , so or because is prime. So then either or is mod . So is an integral domain when is prime.
The integral domain condition is weaker than the field condition:
Every field is an integral domain, but not every integral domain is a field.
First there is a Lemma: For all elements of a ring , .
Proof of lemma: Since is the additive identity, . Then by the distributive law. But we can add the additive inverse of to both sides, to get . The proof of is similar.
Now for the proof of the result. If every nonzero element has a multiplicative inverse, suppose but and are nonzero. Then multiply both sides by to get so contradiction. So there are no zero-divisors.
To see that not every integral domain is a field, simply note that is an example of an integral domain that is not a field (since e.g. does not have a multiplicative inverse in ).
An ideal of a ring is prime if and or .
An ideal of a ring is maximal if but any ideal that strictly contains is the entire ring . (That is, for an ideal , implies or .)
The ideal of is prime, because if , then , so or (because is a prime number), so or .
It is also maximal, because if is an ideal strictly containing , then there is an element that is not a multiple of . Now, since , there are such that by Bezout's identity, but and are both in , so their sum is, so .
But then for any , is in , so .
On the other hand, is neither prime nor maximal, because but ; and the ideal is strictly larger than but is not the entire ring.
Here is a nice theorem that ties together some of the concepts from this wiki.
Let be a commutative ring, and let be an ideal not equal to . Then:
(1) is an integral domain if and only if is prime
(2) is a field if and only if is maximal.
(1) comes directly from the definitions: if is an integral domain and , then in , so or , so or , so is prime. The converse is similar.
For (2), suppose is maximal; then take a nonzero element . Then is an ideal, and it's strictly bigger than since it contains . So it must equal the whole ring , and in particular it contains .
So there exist such that , and in this becomes so has a multiplicative inverse in . This shows that is a field. The converse is similar.
Note that this immediately shows that every maximal ideal is prime, by the result from the previous section.
A nonempty set in a ring is called an ideal if
(1) it is closed under addition:
(2) it "swallows up" under multiplication: .
A proper ideal is one that is not equal to the entire ring.
A proper ideal is prime if implies that or .
A proper ideal is maximal if there are no ideals in between it and the entire ring: if is an ideal, then implies or .
How many ideals of are prime but not maximal?