2.4. Rings

So far we have studied algebraic structures with only one operation. Now we study rings which are sets with two (compatible) binary operations. Unlike groups, these two operations are usually denoted by + and · . One can, of course, go for general notations for these operations. However, that generalization doesn’t seem to pay much, but complicates matters. We stick to the conventions.

2.4.1. Definition and Basic Properties

Definition 2.12.

A ring (R, +, ·) (or R in short) is a set R together with two binary operations + and · on R such that the following conditions are satisfied. As in the case of multiplicative groups we write ab for a · b.

  1. Additive group The set R is an Abelian group under +. The additive identity is denoted by 0.

  2. · is associative (ab)c = a(bc) for every a, b, .

  3. · is commutative ab = ba for every a, .

  4. Multiplicative identity There is an element (denoted by 1) in R such that a · 1 = 1 · a = a for every . The element 1 is called the identity of R.

  5. Distributivity The operation · is distributive over +, that is, a(b+c) = ab + ac and (a + b)c = ac + bc for every a, b, .

Notice that it is more conventional to define a ring as an algebraic structure (R, +, ·) that satisfies conditions (1), (2) and (5) only. A ring (by the conventional definition) is called a commutative ring (resp. a ring with identity), if it (additionally) satisfies condition (3) (resp. (4)). As per our definition, a ring is always a commutative ring with identity. Rings that are not commutative or that do not contain the identity element are not used in the rest of the book. So let us be happy with our unconventional definition of a ring.[3]

[3] Cool! But what’s circular in a ring? Historically, such algebraic structures were introduced by Hilbert to designate a Zahlring (a number ring, see Section 2.13). If α is an algebraic integer (Definition 2.95) and we take a Zahlring of the form and consider the powers α, α2, α3, . . . , we eventually get an αd which can be expressed as a linear combination of the previous (that is, smaller) powers of α. This is perhaps the reason that prompted Hilbert to call such structures “rings”. Also see Footnote 1.

We do not rule out the possibility that 0 = 1 in R. In that case, for any , we have a = a · 1 = a · 0 = 0 (See Proposition 2.6), that is to say, the set R consists of the single element 0. In this case, R is called the zero ring and is denoted (by an abuse of notation) by 0.

Finally, note that R is, in general, not a group under multiplication. This is because we do not expect a ring R to contain the multiplicative inverse of every element of R. Indeed the multiplicative inverse of the element 0 exists if and only if R = 0.

Example 2.6.

  1. The sets , , and are all rings under usual addition and multiplication. Each of , and contains the multiplicative inverse of every non-zero element, whereas the only elements in , that have multiplicative inverses, are ±1.

  2. Let denote the set {0, 1, . . . , n – 1} for an integer n ≥ 2. Then is a ring under addition and multiplication modulo n. The additive identity is 0 and the multiplicative identity is 1. Later we see a more formal definition of this ring. Recall from Example 2.1 how we have defined the groups and under addition and multiplication modulo n. These groups have a connection with the ring as we will shortly see.

  3. Let R be a ring and S a set. The set of all functions S → R is a ring under pointwise addition and multiplication of functions (that is, if f and g are two such functions, then we define (f + g)(a) := f(a) + g(a) and (f g)(a) := f(a)g(a) for every ). The additive (resp. multiplicative) identity in this ring is the constant function 0 (resp. 1).

  4. Let R be a ring. The set R[X] of all polynomials in one indeterminate X and with coefficients from R is a ring. The identity elements in R[X] are the constant polynomials 0 and 1. The addition and multiplication operations in R[X] are the standard ones on polynomials. For a non-zero polynomial , the largest non-negative integer d for which the coefficient of Xd is non-zero is called the degree of the polynomial f and is denoted by deg f. The coefficient of Xdeg f in f is called the leading coefficient of f and is denoted by lc(f). The degree of the zero polynomial is conventionally taken to be –∞. A non-zero polynomial with leading coefficient 1 is called a monic polynomial.

    More generally, for one can define the ring R[X1, . . . , Xn] of multivariate polynomials over R. Polynomial rings are of paramount importance in algebra and number theory. We devote Section 2.6 to a study of these rings.

    We also define the ring R(X) of rational functions over R, which consists of elements of the form f/g with f, , g ≠ 0. More generally, the set of elements f/g with f, , g ≠ 0, is a ring denoted R(X1, . . . , Xn).

  5. Let Ri, , be a family of rings, and the product of the sets Ri, , that is, the set of all ordered tuples indexed by I. For tuples and , define the sum and the product . It is easy to see that R is a ring with identity elements and . It is called the direct product of the rings Ri, . If I is of finite cardinality n and if Ri = A for all , then is denoted in short by An.

Proposition 2.6.

Let R be a ring. For all a, , we have:

  1. a · 0 = 0 · a = 0

  2. a(–b) = (–a)b = –ab

  3. (–a)(–b) = ab

Proof

  1. a · 0 = a · (0 + 0) = a · 0 + a · 0, so that a · 0 = 0. Similarly, 0 · a = 0.

  2. By (1), 0 = a · 0 = a(b + (–b)) = ab + a(–b), that is, a(–b) = –ab. Similarly, (–a)b = –ab.

  3. (–a)(–b) = –(a(–b)) = –(–ab) = ab.

Definition 2.13.

Let R be a ring.

  1. An element is called a zero-divisor of R, if ab = 0 for some , b ≠ 0. By this definition, 0 is a zero-divisor of R, unless R = 0. The elements 0, 3, 5, 6, 9, 10 and 12 are all the zero-divisors of .

  2. An element is called a unit of R, if there exists an element such that ab = 1. The elements 1 and –1 are units in any ring. It is easy to see that an element cannot be simultaneously a zero-divisor and a unit. The set of all units in a ring R is denoted by R* and is a group under the multiplication of the ring R (See Exercise 2.21), called the multiplicative group or the group of units of R. The multiplicative group of the ring (Example 2.6) is .

  3. An element is called nilpotent, if ak = 0 for some . By this definition, 0 is a nilpotent element in any ring. It is also evident that every nilpotent element in a non-zero ring is a zero-divisor. An example of a non-zero nilpotent element in a ring is .

  4. An element is called idempotent, if a2 = a. In every ring, 0 and 1 are idempotent. The element 6 is idempotent in . It is easy to check that 0 is the only element in a ring, that is both nilpotent and idempotent.

Definition 2.14.

Let R be a ring.

  1. R is called an integral domain (or simply a domain), if R ≠ 0 and if R contains no non-zero zero-divisors. Examples of integral domains: , , , , . On the other hand, 3 · 5 = 0 in , so Z15 is not an integral domain.

  2. R is called a field, if R ≠ 0 and if R* = R {0}, that is, if every non-zero element of R is a unit. This means that in a field one can divide any element by any non-zero element. The most common fields are , and . Note that is not a field, since, for example, 2 does not have a multiplicative inverse in .

  3. A field R with #R finite is called a finite field. The simplest examples of finite fields are the fields for prime integers p. In fact, it is easy to see that is a field if and only if n is a prime. Finite fields are widely applied for building various cryptographic protocols. See Section 2.9 for a detailed study of finite fields.

Corollary 2.1.

A field is an integral domain.

Proof

Recall from Definition 2.13 that an element in a ring cannot be simultaneously a unit and a zero-divisor.

Definition 2.15.

Let R be a non-zero ring. The characteristic of R, denoted char R, is the smallest positive integer n such that 1 + 1 + · · · + 1 (n times) = 0. If no such integer exists, then we take char R = 0.

, , and are rings of characteristic zero. If R is a non-zero finite ring, then the elements 1, 1 + 1, 1 + 1 + 1, · · · cannot be all distinct. This shows that there are positive integers m and n, m < n, such that 1+1+· · · + 1 (n times) = 1 + 1 + · · · + 1 (m times). But then 1 + 1 + · · · + 1 (n – m times) = 0. Thus any non-zero finite ring has positive (that is, non-zero) characteristic. If char R = t is finite, then for any one has .

In what follows, we will often denote by n the element 1 + 1 + · · · + 1 (n times) of any ring. One should not confuse this with the integer n. One can similarly identify a negative integer –n with the ring element –(1 + 1 + · · · + 1)(n times) = (–1) + (–1) + · · · + (–1)(n times).

Proposition 2.7.

Let R be an integral domain of positive characteristic p. Then p is a prime.

Proof

If p is composite, then we can write p = mn with 1 < m < p and 1 < n < p. But then p = mn = 0 (in R). Since R is an integral domain, we must have m = 0 or n = 0 (in R). This contradicts the minimality of p.

2.4.2. Subrings, Ideals and Quotient Rings

Just as we studied subgroups of groups, it is now time to study subrings of rings. It, however, turns out that subrings are not that important for the study of rings as the subsets called ideals are. In fact, ideals (and not subrings) help us construct quotient rings. This does not mean that ideals are “normal” subrings! In fact, ideals are, in general, not subrings at all, and conversely. The formal definitions are waiting!

Definition 2.16.

Let R be a ring. A subset S of R is called a subring of R, if S is a ring under the ring operations of R. In this case, one calls R a superring or a ring extension of S.

If R and S are both fields, then S is often called a subfield of R and R a field extension (or simply an extension) of S. In that case, one also says that S ⊆ R is a field extension or that R is an extension over S.

is a subring of , and , whereas and are field extensions.

We demand that a ring always contains the multiplicative identity (Definition 2.12). This implies that if S is a subring of R, then for all integers n, the elements are also in S (though they need not be pairwise distinct). Similarly, if R and S are fields, then S contains all the elements of the form mn–1 for m, , (cf. Exercise 2.26). Thus , the set of all even integers, is not a subring of , though it is a subgroup of (, +) (Example 2.2).

Definition 2.17.

Let R be a ring. A subset of R is called an ideal of R, if is an additive subgroup of (R, +) and if for all and .[4]

[4] Kummer introduced the concept of ideal numbers. Later Dedekind reformulated Kummer’s notion of ideal numbers to define what we now know as ideals.

In this book, we will use Gothic letters (usually lower case) like , , , , to denote ideals.[5]

[5] Mathematicians always run out of symbols. Many believe if it is Gothic, it is just ideal!

The condition for being an ideal is in one sense more stringent than that for being a subring, that is, an ideal has to be closed under multiplication by any element of the entire ring. On the other hand, we do not demand an ideal to necessarily contain the identity element 1. In fact, is an ideal of . Conversely, is a subring of but not an ideal. Subrings and ideals are different things.

Example 2.7.

  1. Let R be any ring. The subset {0} is an ideal of R, called the zero ideal and denoted also by 0. Similarly, the entire ring R is an ideal of R and is called the unit ideal. Note that if an ideal contains a unit u of R, then 1 = u–1u is also in and so for every . It follows that an ideal of R is the unit ideal if and only if contains a unit—a justification for the name.

  2. The integral multiples of an integer n form an ideal of denoted by . More generally, for any ring R and for any , the set is an ideal of R and is denoted by Ra or aR or 〈a〉. Such an ideal is called a principal ideal. (See also Definition 2.18.)

  3. Let R be a ring and let , , be a family of ideals of R. The intersection is an ideal of R. The set of finite sums the form (where and ) is an ideal of R. It is called the sum of the ideals , , and is denoted by . The union is, in general, not an ideal of R. In fact, the sum is the smallest ideal that contains (the set) .

Proposition 2.8.

The only ideals of a field are the zero ideal and the unit ideal.

Proof

By definition, every non-zero element of a field is a unit.

Definition 2.18.

Let R be a ring and ai, , a family of elements of R. The ideal generated by ai, , is defined to be the sum of the principal ideals Rai. We denote this as . In this case, we also say that is generated by ai, . If I is finite, then we say that is finitely generated. In particular, if #I = 1, then is a principal ideal (See Example 2.7).

An integral domain every ideal of which is principal is called a principal ideal domain or PID in short. A ring every ideal of which is finitely generated is called Noetherian. Thus principal ideal domains are Noetherian.

Note that an ideal may have different generating sets of varying cardinalities. For example, the unit ideal in any ring is principal, since it is generated by 1. The integers 2 and 3 generate the unit ideal of , since . However, neither 2 nor 3 individually generates the unit ideal of . Indeed, using Bézout’s relation (Proposition 2.16) one can show that for every there is a (minimal) generating set of the unit ideal of , that contains exactly n integers. Interested readers may try to construct such generating sets as an (easy) exercise.

Theorem 2.6.

is a principal ideal domain.

Proof

The zero ideal is generated by 0. Let be a non-zero ideal of and let a be the smallest positive integer contained in . We claim that . Clearly, . For the converse, take . We can write b = aq + r, where q and r are the quotient and the remainder of (Euclidean) division of b by a. Now and since 0 ≤ r < a, by the choice of a we must have r = 0, so that .

A very similar argument proves the following theorem. The details are left to the reader. Also see Exercise 2.31.

Theorem 2.7.

If K is a field, then K[X] is a principal ideal domain.

We now prove a very important theorem:

Theorem 2.8. Hilbert’s basis theorem

If R is a Noetherian ring, then so is the polynomial ring R[X1, . . . , Xn] for . In particular, the polynomial rings and K[X1, . . . , Xn] are Noetherian, where K is a field.

Proof

Using induction on n we can reduce to the case n = 1. So we prove that if R is Noetherian, then R[X] is also Noetherian. Let be a non-zero ideal of R[X]. Assume that is not finitely generated. Then we can inductively choose non-zero polynomials f1, f2, f3, · · · from such that for each the polynomial fi is one having the smallest degree in . Let di := deg fi. Then d1 ≤ d2 ≤ d3 ≤ · · ·. Let ai denote the leading coefficient of fi. Consider the ideal in R. By hypothesis, is finitely generated, say, . This, in particular, implies that for some . But then the polynomial belongs to , is non-zero and has degree < dr+1, a contradiction to the choice of fr+1. Thus must be finitely generated.

Two particular types of ideals are very important in algebra.

Definition 2.19.

Let R be a ring.

  1. An ideal of R is called a prime ideal, if and if implies or for a, . The second condition is equivalent to saying that if and , then the product . For a prime integer p, the principal ideal of is prime. On the other hand, for a composite integer n the ideal of is not prime. For example, and , but the product .

  2. An ideal of R is called a maximal ideal, if and if for any ideal satisfying we have or . This means that there are no non-unit ideals of R properly containing . All the ideals of for prime integers p are maximal ideals (Corollary 2.3). Next consider the polynomial ring and the principal ideal 〈X〉 of R. It is easy to see that 〈X〉  〈X, 2〉  R. Thus 〈X〉 is not maximal.

Prime and maximal ideals can be characterized by some nice equivalent criteria. See Proposition 2.9.

Definition 2.20.

Let R be a ring and an ideal of R. Then is a subgroup of the group (R, +). Since (R, +) is Abelian, is a normal subgroup (Definition 2.6). Thus the cosets , , form an additive Abelian group. We define multiplication on these cosets as . It is easy to check that this multiplication is well-defined. Furthermore, the set of these cosets, denoted , becomes a ring under this addition and multiplication. The ring is called the quotient ring of R with respect to .

We say that two elements a, are congruent modulo an ideal (of R) and write a ≡ b (mod ), if . Thus a ≡ b (mod ) if and only if a and b lie in the same coset of , that is, .

Example 2.8.

  1. For any ring R, the quotient ring R/0 is essentially the same as R and the quotient ring R/R is the zero ring.

  2. The ring of Example 2.6 is formally defined to be the quotient ring . Convince yourself that both these definitions are equivalent.

Proposition 2.9.

Let R be a ring and an ideal of R.

  1. is a prime ideal of R if and only if is an integral domain.

  2. is a maximal ideal of R if and only if is a field.

Proof

  1. Let a, be arbitrary. Then is prime ⇔ implies or implies or is an integral domain.

  2. Let be a maximal ideal. Choose . Then . Consider the ideal . Since is maximal, we must have . This means that a + cb = 1 for some and . Then which implies that is a unit in . That is, is a field.

    Conversely, let be a field. Consider any ideal of R with . Choose any . Then . By hypothesis, there exists such that , that is, . Hence , that is, .

The last proposition in conjunction with Corollary 2.1 indicates:

Corollary 2.2.

Maximal ideals are prime.

Corollary 2.3.

For every , the quotient ring is a field. In particular, is a maximal ideal of .

Proof

Since is a prime ideal of , is an integral domain. But is finite, so by Exercise 2.25 is a field.

2.4.3. Homomorphisms

Recall how we have defined homomorphisms of groups. In a similar manner, we define homomorphisms of rings. A ring homomorphism is a map from one ring to another, which respects addition, multiplication and the identity element. More precisely:

Definition 2.21.

Let R and S be rings. A map f : R → S is called a (ring)homomorphism, if f(a+b) = f(a) + f(b) and f(ab) = f(a)f(b) for all a, and if f(1) = 1. A homomorphism f : R → S is called an isomorphism, if there exists a homomorphism g : S → R such that g ο f = idR and f ο g = idS. As in the case of groups, bijectivity of f as a function is both necessary and sufficient for a homomorphism f : R → S to be an isomorphism. If f : R → S is an isomorphism, we write R ≅ S and say that R is isomorphic to S or that R and S are isomorphic.

A homomorphism f : R → R is called an endomorphism of R. An automorphism is a bijective endomorphism.

Example 2.9.

  1. For any ring extension R ⊆ S, the canonical inclusion a ↦ a is a homomorphism from R → S. In particular, the identity map on any ring is an automorphism.

  2. Let R be a ring and an ideal of R. The canonical surjection that takes is a ring homomorphism.

  3. Let R be a ring and let . The map R[X] → R that takes f(X) ↦ f(a) is a ring homomorphism and is called the substitution homomorphism.

  4. The map taking n ↦ –n is not a ring homomorphism, since it maps 1 to –1 (and does not satisfy f(ab) = f(a)f(b) for all a, ).

  5. The map that maps z = a + ib to its conjugate is an automorphism of the field .

Proposition 2.10.

Let f : R → S be a ring homomorphism.

  1. If is a unit, then f(a) is a unit in S and f(a–1) = (f(a))–1.

  2. Let be an ideal in S. Then is an ideal in R. If is prime, then is also prime.

Proof

  1. If ab = 1, then f(a)f(b) = f(ab) = f(1) = 1.

  2. For , a, and b, with f(a) = b and f(a′) = b′, we have and . Thus is an ideal of R. If , then . If is prime (in which case and are proper ideals of R and S respectively), then or . But then or .

The ideal of the above proposition is called the contraction of and is often denoted by . If R ⊆ S and f is the inclusion homomorphism, then .

Definition 2.22.

Let f : R → S be a ring homomorphism. The set is called the kernel of f and is denoted by Ker f. The set is called the image of f and is denoted by f(R) or Im f.

Theorem 2.9. Isomorphism theorem

With the notations of the last definition, Ker f is an ideal of R, Im f is a subring of S and R/ Ker f ≅ Im f.

Proof

Consider the map that takes a + Ker f ↦ f(a). It is easy to verify that is a well-defined ring homomorphism and is bijective. The details are left to the reader. Also see Theorem 2.3.

Definition 2.23.

Two ideals and of a ring R are called relatively prime or coprime if , that is, if there exist and with a + b = 1.

Theorem 2.10. Chinese remainder theorem (CRT)

Let R be a ring and . Let be ideals in R such that for all i, j, i ≠ j, the ideals and are relatively prime. Then is isomorphic to the direct product .

Proof

The assertion is obvious for n = 1. So assume that n ≥ 2 and define the map by for all . Since for all i, the map is well-defined. It is easy to see that is a ring homomorphism. In order to show that is injective, we let . This means that , that is, for all i. Then , that is, . The trickier part is to prove that is surjective. Let . Let us consider the ideal for each i. For a given i, there exist for each j ≠ i elements and with αj + βj = 1. Multiplying these equations shows that we have a such that γi + δi = 1, where . (This shows that for all i.) Now consider the element . It follows that for all i, that is, .

In Section 2.5, we will see an interesting application of this theorem. Notice that the injectivity of in the last proof does not require the coprimality of ; the surjectivity of requires this condition.

2.4.4. Factorization in Rings

Now we introduce the concept of divisibility in a ring. We also discuss about an important type of rings known as unique factorization domains. This study is a natural generalization of that of the rings and K[X], K a field.

Definition 2.24.

Let R be a ring, a, and . Also let K be a field.

  1. We say that a divides b and write a|b, if there exists an element such that b = ac. If a does not divide b, we write ab. In , for example, –31|899, since 899 = (–31) · (–29). By this definition, any element divides 0, whereas 0 divides no element other than 0.

  2. It is easy to see that a|b and b|a if and only if b = ca for some unit . In that case, we say that a and b are associates of each other. The relation of being associate is an equivalence relation on R (or R {0}), as can be easily verified. The only associates of , a ≠ 0, are ±a, since ±1 are the only units in . Two non-zero polynomials f and g of K[X] are associates if and only if f = αg for some .

  3. A non-zero non-unit is called a prime, if p|ab implies either p|a or p|b. One can check easily that p is prime if and only if the principal ideal 〈p〉 = pR is a prime ideal.

  4. A non-zero non-unit is called irreducible, if p = ab implies either a or b is a unit.

Note that for the concepts of prime and irreducible elements are the same. This is indeed true for any PID (Proposition 2.12). Thus our conventional definition of a prime integer p > 0 as one which has only 1 and p as (positive) divisors tallies with the definition of irreducible elements above. For the ring K[X], on the other hand, it is more customary to talk about irreducible polynomials instead of prime polynomials; they are the same thing anyway.

Proposition 2.11.

Let R be an integral domain and a prime. Then p is irreducible.

Proof

Let p = ab. Then p|(ab), so that by hypothesis p|a or p|b. If p|a, then a = up for some . Hence p = ab = upb, that is, (1 – ub)p = 0. Since R is an integral domain and p ≠ 0, we have 1 – ub = 0, that is, ub = 1, that is, b is a unit. Similarly, p|b implies a is a unit.

Proposition 2.12.

Let R be a PID. An element is prime if and only if p is irreducible.

Proof

[if] Let p be irreducible, but not prime. Then there are a, such that a ∉ 〈p〉 and b ∉ 〈p〉, but . Consider the ideal . Since , we have p = cα for some . By hypothesis, p is irreducible, so that either c or α is a unit. If c is a unit, 〈p〉 = 〈α〉 = 〈p〉 + 〈a〉, that is, , a contradiction. So α is a unit. Then 〈p〉 + 〈a〉 = R which implies that there are elements u, such that up + va = 1. Similarly, there are elements u′, such that u′p + v′b = 1. Multiplying these two equations gives (uu′p + uv′b + u′va)p + (vv′)ab = 1. Now , so that ab = wp for some . But then (uu′p + uv′b + u′va + vv′w)p = 1, which shows that p is a unit, a contradiction.

[only if] Immediate from Proposition 2.11.

Definition 2.25.

An integral domain R is called a unique factorization domain or a UFD in short, if every non-zero element can be written as a product a = up1 · · · pr, where , and p1, . . . , pr are prime elements (not necessarily distinct) of R. Moreover, such a factorization is unique up to permutation of the primes p1, . . . , pr and up to multiplication of the primes by units. This factorization can also be written as , where , , q1, . . . , qs are pairwise non-associate primes and αi > 0 for i = 1, . . . , s. Some authors also use the term factorial ring or factorial domain in order to describe a UFD.

If is a prime and , a ≠ 0, then the multiplicity of p in a is the nonnegative integer v such that pv|a, but pv+1a. This integer v is denoted by vp(a). It is clear form the definition that for every , a ≠ 0, there exist only finitely many non-associate primes p for which vp(a) > 0.

Proposition 2.13.

Let R be a UFD. An element is prime if and only if p is irreducible.

Proof

The only if part is immediate from Proposition 2.11. For proving the if part, let p = up1 · · · pr ( and pi primes in R) be irreducible. If r = 0, p is a unit, a contradiction. If r > 1, then p can be written as the product of two non-units up1 · · · pr–1 and pr, again a contradiction. So r = 1.

A classical example of an integral domain that is not a UFD is . In this ring, we have two essentially different factorizations of 6 into irreducible elements. The failure of irreducible elements to be primes in such rings is a serious thing to patch up!

Theorem 2.11.

A PID is a UFD

Proof

Let R be a PID and . We show that a has a factorization of the form a = up1 · · · pr, where u is a unit and p1, . . . , pr are prime elements of R. If a is a unit, we are done. So assume that a =: a0 is a non-unit and let . Since , there is a maximal ideal containing (Exercise 2.23). Then p1 is a prime that divides a0. Let a0 = a1p1. We have . If is the unit ideal, we are done. Otherwise we choose as before a prime p2 dividing a1 and with a1 = a2p2 get the ideal properly containing . Repeating this process we can generate a strictly ascending chain of ideals of R. Since R is a PID and hence Noetherian, this process must stop after finitely many steps (Exercise 2.33).

The converse of the above theorem is not necessarily true. For example, the polynomial ring K[X1, . . . , Xn] over a field K is a UFD for every , but not a PID for n ≥ 2.

Divisibility in a UFD can be rephrased in terms of prime factorizations. Let R be a UFD and let the non-zero elements a, have the prime factorizations and with units u, u′, pairwise non-associate primes p1, . . . , pr and with αi ≥ 0 and βi ≥ 0. Then a|b if and only if αi ≤ βi for all i = 1, . . . , r. This notion leads to the following definitions.

Definition 2.26.

Let R be a UFD and let a, have prime factorizations as in the last paragraph. Any associate of , is called a greatest common divisor of a and b and is denoted by gcd(a, b). Clearly, gcd(a, b) is unique up to multiplication by units of R. Similarly, any associate of , is called a least common multiple of a and b and is denoted by lcm(a, b). lcm(a, b) is again unique up to multiplication by units of R. The gcd of a ≠ 0 and 0 is taken to be an associate of a, whereas gcd(0, 0) is undefined. On the other hand, lcm(a, 0) is defined to be 0 for any .

It is clear that these definitions of gcd and lcm can be readily generalized for any arbitrary finite number of elements.

Corollary 2.4.

Let R be a UFD and a, not both zero. Then gcd(a, b) · lcm(a, b) is an associate of ab.

Proof

Immediate from the definitions.

Corollary 2.5.

Let R be a UFD and a, b, with a|bc. If gcd(a, c) = 1, then a|b.

Proof

Consider the prime factorizations of a, b and c.

For a PID, the gcd and lcm have equivalent characterizations.

Proposition 2.14.

Let R be a PID and a, b be non-zero elements of R. Let d be a gcd of a and b. Then 〈d〉 = 〈a〉 + 〈b〉. If f is an lcm of a and b, then 〈f〉 = 〈a〉 ∩ 〈b〉.

Proof

Let 〈a〉 + 〈b〉 = 〈c〉. We show that c and d are associates. There exist u, such that ua + vb = c. Since d|a and d|b, we have d|c. On the other hand, , so that c|a. Similarly c|b. Considering the prime factorizations of a and b one can then readily verify that c|d. The proof for the second part is similar and is left to the reader.

A direct corollary to the last proposition is the following.

Corollary 2.6.

Let R be a PID, a, (not both zero) and d a gcd of a and b. Then there are elements u, such that ua + vb = d. In particular, the ideals 〈a〉 and 〈b〉 are relatively prime if and only if gcd(a, b) is a unit. In that case, we also say that the elements a and b are relatively prime or coprime.

This completes our short survey of factorization in rings. Note that and K[X] (for a field K) are PID and hence UFD. Thus all the results we have proved in this section apply equally well to both these rings. It is because of this (and not of a mere coincidence) that these two rings enjoy many common properties. Thus our abstract treatment saves us from the duplicate effort of proving the same results once for integers (Section 2.5) and once more for polynomials (Section 2.6).

Exercise Set 2.4

2.21For a non-zero ring R, prove the following assertions:

  1. A unit of R is not a zero-divisor.

  2. The product of two units of R is again a unit.

  3. The product of two non-units of R is again a non-unit.

  4. The element 0 is not a unit in R.

  5. The element 1 is always a unit in R.

  6. If a is a unit and ab = ac, then b = c.

Let K be a field. What are the units in the polynomial ring K[X]? In K[X1, . . . , Xn]? In the ring K(X) of rational functions? In K(X1, . . . , Xn)?

2.22

Binomial theorem Let R be a ring, a, and . Show that

where

are the binomial coefficients.

2.23Show that every non-zero ring has a maximal (and hence prime) ideal. More generally, show that every non-unit ideal of a non-zero ring is contained in a maximal ideal. [H]
2.24Let R be a ring.

  1. Show that the set of all nilpotent elements of R is an ideal of R. This ideal is usually denoted by and is called the nilradical of R.

  2. Show that the quotient ring has no non-zero nilpotent elements. (The ring is called the reduction of R and is often written as Rred. If , then we say that R is reduced. Thus is always reduced.)

  3. Show that the nilradical of R is the intersection of the prime ideals of R. [H]

2.25Show that a finite integral domain R is a field. [H]
2.26Let R be a ring of characteristic 0. Show that:

  1. R contains infinitely many elements.

  2. If R is an integral domain, then R contains as subring an isomorphic copy of .

  3. If R is a field, then R contains as subfield an isomorphic copy of .

2.27Let f : R → S be a ring-homomorphism and let and be ideals in R and S respectively. Find examples to corroborate the following statements.

  1. Let be such that f(a) is a unit in S. Then a need not be a unit in R.

  2. The set need not be an ideal of S.

  3. If and if is maximal, then need not be maximal.

2.28Let K be a field.

  1. Show that a homomorphism from K to any non-zero ring is injective.

  2. Let L be another field and let f : K → L and g : L → K be homomorphisms such that g ο f = idK. Show that f and g are isomorphisms.

2.29

  1. Show that a ring R is an integral domain if and only if 0 is a prime ideal of R.

  2. Give an example of a reduced ring that is not an integral domain. (Note that an integral domain is always reduced.)

2.30Let R be a ring and let and be ideals of R with . Show that is an ideal of and that . [H]
2.31An integral domain R is called a Euclidean domain (ED) if there is a map satisfying the following two conditions:

  1. ν(a) ≤ ν(ab) for all a, .

  2. For every a, with b ≠ 0, there exist (not necessarily unique) q, such that a = qb + r with r = 0 or ν(r) < ν(b).

Show that:

  1. is a Euclidean domain with ν(a) = |a| for a ≠ 0.

  2. The polynomial ring K[X] over a field K is a Euclidean domain with ν(a) = deg a for a ≠ 0.

  3. For d = –2, –1, 2, 3, the ring

    is a Euclidean domain with , a, , not both 0.

  4. A Euclidean domain is a PID (and hence a UFD).

2.32Let R be a ring and an ideal. Consider the set

Show that is an ideal of R. It is called the radical or root of . If , then is called a radical or a root ideal. For arbitrary ideals and of R, prove the following assertions.

  1. .

  2. .

  3. If , then .

  4. If is a prime ideal, then .

  5. if and only if .

  6. .

  7. .

  8. The nilradical .

2.33Let R be a ring. An ascending chain of ideals is a sequence . The ascending chain is called stationary, if there is some such that for all n ≥ n0. Show that the following conditions are equivalent. [H]

  1. R is Noetherian (that is, every ideal of R is finitely generated).

  2. Every ascending chain of ideals in R is stationary.

  3. Every non-empty set of ideals of R has a maximal element.

2.34

  1. Let R be an integral domain. Define the set . Define a relation ~ on S as (a, b) ~ (c, d) if and only if ad = bc. Show that ~ is an equivalence relation on S. Let us denote the equivalence class of by a/b and the set of all equivalence classes of S under ~ by K.

  2. Now define (a/b)+(c/d) := (ad+bc)/(bd) and (a/b)·(c/d) := (ac)/(bd). Show that these definitions make K a field. This field is called the quotient field of R and is denoted as Q(R). This process resembles the formation of rational numbers from the integers. Indeed, .

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