Commutative Rings
Commutative ring theory has its origins in number theory and algebraic geometry in
the 19th century. Today it is of particular importance in algebraic geometry, and
there has been an interesting interaction of algebraic geometry and number
theory, using the methods of commutative algebra. Here we can do no more than
describe the basic techniques and take the first steps in the subject. In Section 10.1
we define the various operations on ideals and use them in Section 10.2 to study
unique factorization. In Section 10.3 we give an account of fractions and examine
the effect of chain conditions in Section IDA. Many rings of algebraic numbers
fail to have unique factorization of elements, but instead have unique factorization
of ideals, and the consequences are studied in Sections 10.5 and 10.6. Sections
10.7-10.10 deal with the properties of rings used in algebraic geometry (but also
of importance in commutative ring theory): equations (Section 10.7), decomposition
of ideals (Section 10.8), dimension (Section 10.9) and the relation between ideals and
algebraic varieties (Section 10.10).
Throughout this chapter all rings will be commutative, unless otherwise stated; for
this reason there is no need to distinguish between left and right modules.
10.1 Operations on Ideals
Historically the first ring to be studied was the ring Z of integers; the term 'ring' was
first used by David Hilbert (1897) in his Zahlbericht for a ring of algebraic integers
(Zahlring). In Z every ideal is principal; in fact ideals were first introduced (by Ernst-
Eduard Kummer) as 'ideal numbers' in rings of algebraic integers which lacked
unique factorization. In Z we can from any two numbers a, b form their highest
common factor (HCF; also called greatest common divisor) (a, b), their product
ab and their least common multiple (LCM) [a, b]. These operations correspond to
operations on ideals in any ring.
We shall denote ideals in a ring R by small gothic letters a, b, C, •..• If a is an
ideal in a commutative ring R, generated by elements al, ... , an> we write
a = (aI, ... , an); then the elements of a are all the linear combinations L riai
(ri E R). In particular, when n = 1, a is principal; thus (a) is the ideal consisting
of all elements ra (r E R).
347
,348 Commutative Rings
Given ideals a, b in R, we define their sum as
a + b = {x + ylx E a,y E b}.
It is easily seen that this is an ideal, the least ideal containing a and b. If a, b are prin-
cipal, say a = (a), b = (b), then their sum, if principal, has the form (d), where d is
an HCF of a and b. For example in Z, (36) + (10) = (2); on the other hand in Z[x],
(x) + (2) = (x, 2) and this cannot be simplified.
Similarly the product is defined as
This operation is again associative and commutative, so that ideals form a com-
mutative monoid under multiplication, with (1) = R as neutral element. If b = a,
the product aa is written a2 and generally an is defined recursively by an = an - I a.
A third operation is the intersection a n b, which corresponds to the LCM when
all ideals are principal.
Let us note the form these definitions take for finitely generated ideals.
If a = (al, . .. , ar ), b = (b l , ... , b,), then a + b = (al, ... , ar , bl , .. . , bs ), while
ab = (alb l , a l b2, ... , albs, a2bl, ... , arb,). This shows in particular that the sum
and product of finitely generated ideals are again finitely generated. By contrast
there is no simple expression for an b, and it need not be finitely generated, even
when both a and b are, but we note that there is a short exact sequence relating
an b to a + b:
A JJ,
o~ an b ----+ a EEl b ----+ a + b ~ o. (10.1.1)
Here f.L is defined as (x, y)f.L = x - y (x E a, y E b) and ker f.L ~ an b; this is easily
verified and will be left to the reader to do.
There is also a form of division for ideals. Given ideals a, b, we define
a: b = {x E Rlxb ~ a}.
The reader will have no difficulty in verifying that this is an ideal. This is so even if b
is an arbitrary subset of R, not necessarily an ideal. Thus for any subset 5 of R we
have
a: 5 = {x E RlxS ~ a}.
The addition, multiplication and intersection of ideals are associative and com-
mutative and satisfy the distributive law: a(b + c) = ab + ac, as is easily checked.
Division is related to the other operations by the formulae:
(a: b)b ~ a~ a: b, (10.1.2)
(n ai) : b = n(ai : b), a: (I: bi) = n(a: bi ), (10.1.3)
(a : b) : c= a : be. (10.1.4)
,10.2 Prime Ideals and Factorization 349
For example, to prove (10.1.4), we have, for any x E R, x E (a: b) : e {} xc ~
a : b {} xbe ~ a {} x E a : be.
The other rules are established similarly.
Exercises
1. Prove the formulae (10.1.2)-(10.1.4).
2. Show that (a n b) (a + b) ~ ab for any ideals a, b in a ring R, and give an example
in Z[x] where the inequality is strict.
3. Show that (a + e)(b + c) ~ ab + e for any ideals a, b, e in a ring R, and give
examples where the inequality is strict.
4. Let a, b be finitely generated ideals in an integral domain. Show that if a + b is
principal, then an b is finitely generated. (Hint. Use the exact sequence (10.1.1).)
5. Let K = k(x, y, Zl, Z2, ... ), where k is a field and x, y, Zi (i = 1,2, ... ) are indeter-
minates. Show that in the k-subalgebra of K generated by x, y, Zi, xy-I zi
(i = 1,2, ... ) neither (x) n (y) nor (x) : (y) is finitely generated.
10.2 Prime Ideals and Factorization
In any integral domain R an element is said to be irreducible or an atom if it is a non-
unit, which cannot be expressed as a product of two non-units. For example, in Z the
atoms are just the prime numbers; Z has the further important property that every
integer can be written as a product of prime numbers in essentially only one way. Let
us define more generally a unique factorization domain (UFD) as an integral domain
in which every non-unit has a complete factorization, i.e. it can be written as a
product of atoms, and in any two complete factorizations of an element the factors
are the same except for the order and unit factors. In order to study UFDs it is con-
venient to single out another property of atoms. By a prime we shall understand an
element p in an integral domain R, not zero or a unit such that for any a, b E R,
plab =} pia or plb. Any prime is necessarily an atom, for if p is a prime and
p = ab (a, b i= 0), then by definition, pia or plb, say pia. Hence a = pc = abc; it
follows that bc = 1, so b is a unit and p is an atom, associated to a (i.e. differing
by a unit factor). We note that the converse, with a finiteness condition, characterizes
UFDs:
Theorem 10.2.1. Let R be an integral domain. Then R is a unique factorization
domain if and only if
(i) every element of R not zero or a unit has a complete factorization and
(ii) every atom is a prime.
Proof. If R is a UFD, then every element not zero or a unit has a complete factor-
ization. Now let p be an atom and suppose that plab, say ab = pc. Take complete
factorizations a = PI ... pn b = ql ... qs' C = ZI ••• Zt. Then PZI ... Zt =
, 350 Commutative Rings
PI ... Prql ... qs and by unique factorization P must be associated to some Pi or qj
and accordingly pia or plb.
Conversely, assume (i), (ii) and take two complete factorizations of an element
(which exist by (i)):
c = PI· .. Pr = ql ... qs· (10.2.1)
We have to show that r = s and after suitable renumbering each qi is associated to Pi.
We shall use induction on r; for r = 1 there is nothing to prove, for then s is also 1.
Now let r > 1; PI is an atom, hence prime and Pllql ... qs, therefore Pllqj for some j,
say j = 1 (by renumbering the %). Since ql is also an atom, it must be associated to
PI> say PI = ql U for a unit u. Dividing (10.2.1) by ql> we obtain
UP2 ... Pr = q2 ... qs·
By induction, r - 1 = s - 1, hence r = s; moreover, we can renumber q2, ... , qr so
that Pi is associated to qi. This also holds for i = 1 and it proves the uniqueness of the
factorization (10.2.1), so R is a UFD. •
Thus a UFD may be characterized as an integral domain in which every element
not zero or a unit can be written as a product of primes. An integral domain satisfy-
ing (i) will be called atomic.
It is easily seen that every Noetherian domain is atomic. For let R be Noetherian
and take a E R. If a is not 0 or a unit, we can split off a non-unit factor bl and con-
tinue in this way: a = blcl = blb2C2 = .... Since the bi are non-units, we have an
ascending chain
this chain must break off, so some Ci must be an atom. We now repeat the process,
taking each bi to be an atom (as we may, by what has just been shown) and so obtain
a factorization of a as a product of atoms. Thus we have
Corollary 10.2.2. Any Noetherian domain is a unique factorization domain if and only
if every atom is a prime. •
Corollary 10.2.3. Every principal ideal domain is a unique factorization domain.
Proof. Let R be a PID; then R is Noetherian, so we need only prove that every atom is
prime. Let a be an atom and suppose that a divides bc but not b, say bc = ah. Let
(a, b) = (d); then d is a common factor of a and b, and so must be a unit, which
may be taken to be 1, i.e. au + bv = 1. Hence c = auc + bcv = a(uc + hv), so alc,
as required. •
For example, the well-known unique factorization of the integers follows from
Corollary 10.2.3 because Z is a PID. This is most easily proved by the division
algorithm, which states that given a, b E Z, where b =f=. 0, there exist q, r E Z such that
a = bq + r, Irl < Ibl. (10.2.2)
Commutative ring theory has its origins in number theory and algebraic geometry in
the 19th century. Today it is of particular importance in algebraic geometry, and
there has been an interesting interaction of algebraic geometry and number
theory, using the methods of commutative algebra. Here we can do no more than
describe the basic techniques and take the first steps in the subject. In Section 10.1
we define the various operations on ideals and use them in Section 10.2 to study
unique factorization. In Section 10.3 we give an account of fractions and examine
the effect of chain conditions in Section IDA. Many rings of algebraic numbers
fail to have unique factorization of elements, but instead have unique factorization
of ideals, and the consequences are studied in Sections 10.5 and 10.6. Sections
10.7-10.10 deal with the properties of rings used in algebraic geometry (but also
of importance in commutative ring theory): equations (Section 10.7), decomposition
of ideals (Section 10.8), dimension (Section 10.9) and the relation between ideals and
algebraic varieties (Section 10.10).
Throughout this chapter all rings will be commutative, unless otherwise stated; for
this reason there is no need to distinguish between left and right modules.
10.1 Operations on Ideals
Historically the first ring to be studied was the ring Z of integers; the term 'ring' was
first used by David Hilbert (1897) in his Zahlbericht for a ring of algebraic integers
(Zahlring). In Z every ideal is principal; in fact ideals were first introduced (by Ernst-
Eduard Kummer) as 'ideal numbers' in rings of algebraic integers which lacked
unique factorization. In Z we can from any two numbers a, b form their highest
common factor (HCF; also called greatest common divisor) (a, b), their product
ab and their least common multiple (LCM) [a, b]. These operations correspond to
operations on ideals in any ring.
We shall denote ideals in a ring R by small gothic letters a, b, C, •..• If a is an
ideal in a commutative ring R, generated by elements al, ... , an> we write
a = (aI, ... , an); then the elements of a are all the linear combinations L riai
(ri E R). In particular, when n = 1, a is principal; thus (a) is the ideal consisting
of all elements ra (r E R).
347
,348 Commutative Rings
Given ideals a, b in R, we define their sum as
a + b = {x + ylx E a,y E b}.
It is easily seen that this is an ideal, the least ideal containing a and b. If a, b are prin-
cipal, say a = (a), b = (b), then their sum, if principal, has the form (d), where d is
an HCF of a and b. For example in Z, (36) + (10) = (2); on the other hand in Z[x],
(x) + (2) = (x, 2) and this cannot be simplified.
Similarly the product is defined as
This operation is again associative and commutative, so that ideals form a com-
mutative monoid under multiplication, with (1) = R as neutral element. If b = a,
the product aa is written a2 and generally an is defined recursively by an = an - I a.
A third operation is the intersection a n b, which corresponds to the LCM when
all ideals are principal.
Let us note the form these definitions take for finitely generated ideals.
If a = (al, . .. , ar ), b = (b l , ... , b,), then a + b = (al, ... , ar , bl , .. . , bs ), while
ab = (alb l , a l b2, ... , albs, a2bl, ... , arb,). This shows in particular that the sum
and product of finitely generated ideals are again finitely generated. By contrast
there is no simple expression for an b, and it need not be finitely generated, even
when both a and b are, but we note that there is a short exact sequence relating
an b to a + b:
A JJ,
o~ an b ----+ a EEl b ----+ a + b ~ o. (10.1.1)
Here f.L is defined as (x, y)f.L = x - y (x E a, y E b) and ker f.L ~ an b; this is easily
verified and will be left to the reader to do.
There is also a form of division for ideals. Given ideals a, b, we define
a: b = {x E Rlxb ~ a}.
The reader will have no difficulty in verifying that this is an ideal. This is so even if b
is an arbitrary subset of R, not necessarily an ideal. Thus for any subset 5 of R we
have
a: 5 = {x E RlxS ~ a}.
The addition, multiplication and intersection of ideals are associative and com-
mutative and satisfy the distributive law: a(b + c) = ab + ac, as is easily checked.
Division is related to the other operations by the formulae:
(a: b)b ~ a~ a: b, (10.1.2)
(n ai) : b = n(ai : b), a: (I: bi) = n(a: bi ), (10.1.3)
(a : b) : c= a : be. (10.1.4)
,10.2 Prime Ideals and Factorization 349
For example, to prove (10.1.4), we have, for any x E R, x E (a: b) : e {} xc ~
a : b {} xbe ~ a {} x E a : be.
The other rules are established similarly.
Exercises
1. Prove the formulae (10.1.2)-(10.1.4).
2. Show that (a n b) (a + b) ~ ab for any ideals a, b in a ring R, and give an example
in Z[x] where the inequality is strict.
3. Show that (a + e)(b + c) ~ ab + e for any ideals a, b, e in a ring R, and give
examples where the inequality is strict.
4. Let a, b be finitely generated ideals in an integral domain. Show that if a + b is
principal, then an b is finitely generated. (Hint. Use the exact sequence (10.1.1).)
5. Let K = k(x, y, Zl, Z2, ... ), where k is a field and x, y, Zi (i = 1,2, ... ) are indeter-
minates. Show that in the k-subalgebra of K generated by x, y, Zi, xy-I zi
(i = 1,2, ... ) neither (x) n (y) nor (x) : (y) is finitely generated.
10.2 Prime Ideals and Factorization
In any integral domain R an element is said to be irreducible or an atom if it is a non-
unit, which cannot be expressed as a product of two non-units. For example, in Z the
atoms are just the prime numbers; Z has the further important property that every
integer can be written as a product of prime numbers in essentially only one way. Let
us define more generally a unique factorization domain (UFD) as an integral domain
in which every non-unit has a complete factorization, i.e. it can be written as a
product of atoms, and in any two complete factorizations of an element the factors
are the same except for the order and unit factors. In order to study UFDs it is con-
venient to single out another property of atoms. By a prime we shall understand an
element p in an integral domain R, not zero or a unit such that for any a, b E R,
plab =} pia or plb. Any prime is necessarily an atom, for if p is a prime and
p = ab (a, b i= 0), then by definition, pia or plb, say pia. Hence a = pc = abc; it
follows that bc = 1, so b is a unit and p is an atom, associated to a (i.e. differing
by a unit factor). We note that the converse, with a finiteness condition, characterizes
UFDs:
Theorem 10.2.1. Let R be an integral domain. Then R is a unique factorization
domain if and only if
(i) every element of R not zero or a unit has a complete factorization and
(ii) every atom is a prime.
Proof. If R is a UFD, then every element not zero or a unit has a complete factor-
ization. Now let p be an atom and suppose that plab, say ab = pc. Take complete
factorizations a = PI ... pn b = ql ... qs' C = ZI ••• Zt. Then PZI ... Zt =
, 350 Commutative Rings
PI ... Prql ... qs and by unique factorization P must be associated to some Pi or qj
and accordingly pia or plb.
Conversely, assume (i), (ii) and take two complete factorizations of an element
(which exist by (i)):
c = PI· .. Pr = ql ... qs· (10.2.1)
We have to show that r = s and after suitable renumbering each qi is associated to Pi.
We shall use induction on r; for r = 1 there is nothing to prove, for then s is also 1.
Now let r > 1; PI is an atom, hence prime and Pllql ... qs, therefore Pllqj for some j,
say j = 1 (by renumbering the %). Since ql is also an atom, it must be associated to
PI> say PI = ql U for a unit u. Dividing (10.2.1) by ql> we obtain
UP2 ... Pr = q2 ... qs·
By induction, r - 1 = s - 1, hence r = s; moreover, we can renumber q2, ... , qr so
that Pi is associated to qi. This also holds for i = 1 and it proves the uniqueness of the
factorization (10.2.1), so R is a UFD. •
Thus a UFD may be characterized as an integral domain in which every element
not zero or a unit can be written as a product of primes. An integral domain satisfy-
ing (i) will be called atomic.
It is easily seen that every Noetherian domain is atomic. For let R be Noetherian
and take a E R. If a is not 0 or a unit, we can split off a non-unit factor bl and con-
tinue in this way: a = blcl = blb2C2 = .... Since the bi are non-units, we have an
ascending chain
this chain must break off, so some Ci must be an atom. We now repeat the process,
taking each bi to be an atom (as we may, by what has just been shown) and so obtain
a factorization of a as a product of atoms. Thus we have
Corollary 10.2.2. Any Noetherian domain is a unique factorization domain if and only
if every atom is a prime. •
Corollary 10.2.3. Every principal ideal domain is a unique factorization domain.
Proof. Let R be a PID; then R is Noetherian, so we need only prove that every atom is
prime. Let a be an atom and suppose that a divides bc but not b, say bc = ah. Let
(a, b) = (d); then d is a common factor of a and b, and so must be a unit, which
may be taken to be 1, i.e. au + bv = 1. Hence c = auc + bcv = a(uc + hv), so alc,
as required. •
For example, the well-known unique factorization of the integers follows from
Corollary 10.2.3 because Z is a PID. This is most easily proved by the division
algorithm, which states that given a, b E Z, where b =f=. 0, there exist q, r E Z such that
a = bq + r, Irl < Ibl. (10.2.2)
