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MATH 431 PART 3: IDEALS, FACTOR RINGS, AND RING HOMOMORPHISMS
1. Cosets, revisited
Consider a ring R and some subring S of R. For all a ∈ R, a + S is the coset of S containing a. Since a ring is
an abelian group under +, we always have a + S = S + a. We define two operations on the cosets of S; for a, b ∈ R
define the sum and product of cosets by
(1)
(a + S) + (b + S) = (a + b) + S
and
(a + S)(b + S) = (ab) + S.
If we think of these rings as groups under +, we proved in MATH 430 that the sum of cosets is well-defined (everything
is happening in a group!). What about the product of these additive cosets? (The product of cosets is well-defined if
for all s1 , s2 ∈ S, (a + s1 )(b + s2 ) = ab + s3 for some s3 ∈ S.) Work with a partner and figure out a condition on
products of elements of R with elements of S that would make the coset product operation well-defined.
Theorem 1. Let S be a subring of the ring R. The product of additive cosets of S defined by (a + S)(b + S) = (ab) + S
is well-defined if and only if
Definition 1. A subring I of a ring R is called a (two-sided) ideal of R if for every a ∈ I and r ∈ R both ar and ra
are in I. Another way to write this condition is
(The way I think about ideals is that they are subrings that have to “absorb” elements of R.)
Ideals for rings are analogous to normal subgroups for groups; for these subrings, the set of cosets has the structure
of a ring with the operations we defined in equation (1).
Theorem 2 (Ideal test). A subset I of a ring R is an ideal of R if:
(1) I is nonempty,
(2) for all a, b ∈ I then a − b ∈ I,
(3) and for all a ∈ I and r ∈ R then ar, ra ∈ I.
Example 1.
(1) The ring R is an ideal of itself, and the single element set {0} is also an ideal of R.
(2) nZ is a subring of Z. Is it an ideal?
(3) Let F be the ring of functions mapping R to R and let C be the subring of all the constant functions. Is
C an ideal of F ?
(4) For F as above, let N be the set of functions f such that f (2) = 0. Is N an ideal in F ?
(5) For F as above, let S be the set of differentiable functions. Is S an ideal in F ?
Theorem 3. The intersection of ideals in a ring R is also an ideal.
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MATH 431 PART 3: IDEALS, FACTOR RINGS, AND RING HOMOMORPHISMS
Theorem 4. Let R be a commutative ring and let a ∈ R. Ia = {x ∈ R | ax = 0} is an ideal of R. (We call Ia the
annihilator of a.)
Proof. Show that Ia is a subring of R and that rx ∈ Ia for all r ∈ R, x ∈ Ia .
2. Factor rings
In the previous section we showed that if I is an ideal then coset addition and multiplication defined by equation
(1) are well-defined. This is similar to the progression we took when we constructed groups of cosets in MATH 430!
Theorem 5. If I is an ideal of a ring R, then the set R/I of additive cosets a + I form a ring under the operations
given by equation (1).
Proof. Since R is an abelian group under + and the addition operation on cosets is the same as it was for additive
groups, we know that the set R/I is an abelian group under coset addition. Thus, we only need to show that coset
multiplication is closed, associative, and that multiplication distributes over addition. Each of these follows from the
corresponding property of R based on the defined multiplication operation on cosets. Therefore, R/I is a ring.
Definition 2. The ring R/I from the previous theorem is the factor ring of R by I.
Theorem 6 (Properties of factor groups). Let R be a ring and let I be an ideal in R so that R/I is a ring.
(1) If R is a field then R/I is either the trivial ring or is isomorphic to R.
(2) R/I is commutative if and only if rs − sr ∈ I for all r, s ∈ R.
(3) If R has a unity and I 6= R then R/I is a ring with unity.
Example 2.
(1) Z/nZ is a factor ring.
(2) Revisit Example 1 part (3). Describe the elements of F/N .
Remark 1. Not all properties of R are inherited by the factor ring R/I! For example, problems 12-14 in the Section
26 exercises ask you to find examples of factor rings of integral domains which are fields and those that contain zero
divisors. Be careful!
3. Ring homomorphisms
Ring homomorphisms have many of the same properties and uses that group homomorphisms had in MATH 430.
Below we state the key properties and definitions carefully.
As review:
Definition 3. Let R and R0 be rings. A map φ : R → R0 is a ring homomorphism if
φ(a + b) = φ(a) + φ(b)
and φ(ab) = φ(a)φ(b)
for all a, b ∈ R. For S ⊆ R, φ[S] = {φ(a) | a ∈ S}. For S 0 ⊆ R0 , φ−1 [S 0 ] = {a ∈ R | φ(a) ∈ S 0 }.
Theorem 7 (Properties of ring homomorphisms). Let φ : R → R0 be a ring homomorphism.
MATH 431 PART 3: IDEALS, FACTOR RINGS, AND RING HOMOMORPHISMS
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(1) If 0 is the additive identity in R then
(2) If a ∈ R then φ(−a) =
(3) If R has a unity 1 then φ(1) is a unity for φ[R].
(4) If S is a subring of R then φ[S]
(5) If S 0 is a subring of R0 then
(6) If I is an ideal of R then φ[I] is an ideal of φ[R].
(7) If I 0 is an ideal of φ[R] or R0 then
Proof. Parts (1)-(3) have been proved for group homomorphisms and still true in this setting. You’ll prove (6) for
homework. For (4), we proved in MATH 430 that φ[S] is a subgroup of R0 , so we only need to show that multiplication
is closed in φ[S].
Definition 4. Let φ : R → R0 be a ring homomorphism. The kernel of φ is
ker φ = φ−1 [00 ] = {r ∈ R | φ(r) = 00 } .
Theorem 8. A ring homomorphism φ : R → R0 is one-to-one if and only if
Theorem 9. For a ring homomorphism φ : R → R0 , ker φ is an ideal of R.
Now we will consider the relationship between ring homomorphisms and factor rings in analogy to our exploration of group homomorphisms and factor groups in MATH 430. In particular, since we know that the kernel of a
homomorphism is always an ideal, R/ ker φ is always a factor ring.
Theorem 10 (The Fundamental Homomorphism Theorem). Let φ : R → R0 be a ring homomorphism and let the
ideal I = ker φ. Then R/ ker φ = R/I is a factor ring and is isomorphic to φ[R] via the map µ : R/I → φ[R] defined
by µ(a + I) = φ(a). Also, the map γ : R → R/I given by γ(x) = x + I is a ring homomorphism with kernel I such
that φ(a) = µγ(a).
Remark 2. This theorem is in direct analogue with the Fundamental Homomorphism Theorem for groups (Thm 6,
Part 7 notes, MATH 430 Fall 2014). It gives us a way to construct factor rings from homomorphisms, and the existence
of a map φ with kernel I for every factor ring R/I.
√
Example 3. What factor ring is constructed using φ√2 : Q[x] → C where φ√2 (f (x)) = f ( 2)?
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MATH 431 PART 3: IDEALS, FACTOR RINGS, AND RING HOMOMORPHISMS
4. Types of ideals: Maximal, prime, and principal
Now that we know what an ideal is, we will define some adjectives to apply to ideals to denote specific properties.
Definition 5. Let R be a nonzero ring.
• R is the improper ideal of R.
• {0} is the trivial ideal of R.
• All other ideals of R are proper nontrivial ideals of R.
Theorem 11. If R is a ring with unity and I is an ideal of R containing a unit then I = R.
Corollary 12. If R is a field, then R contains
What does this corollary imply about factor rings of fields?
We have seen examples of factor rings which are fields, integral domains, or even have zero divisors. How does the
choice of ideal influence these possibilities? Are there properties of ideals that tell us when our factor ring has certain
properties?
Definition 6. A maximal ideal of a ring R is an ideal M 6= R such that there is no proper ideal I of R properly
containing M .
Example 4. The subring of Z[x] consisting of polynomials with a zero constant term is an ideal. Is it maximal?
Example 5. pZ is a maximal ideal of Z for any prime number p. What do we know about Z/pZ?
Theorem 13. Let R be a commutative ring with unity. Then M is a maximal ideal of R if and only if R/M is a
Corollary 14. A commutative ring with unity is a field if and only if it has no proper nontrivial ideals.
Fields are awesome, so it is nice to know how to construct them as factor rings. But sometimes we don’t need all
the properties that come with a field; R/I is an integral domain if and only if it has no zero divisors. In the language
of cosets, this means that (a + I)(b + I) = 0 + I = I if and only if
Definition 7. An ideal I 6= R in a commutative ring R is a prime ideal if
Example 6. pZ is a prime ideal in Z, and {0} is a prime ideal in any integral domain.
Theorem 15. Let R be a commutative ring with unity and let I 6= R be an ideal of R. Then R/I is an integral
domain if and only if I is a prime ideal in R.
All fields are integral domains, but not all integral domains are fields. What does that imply about the relationship
between maximal ideals and prime ideals?
Corollary 16. Let R be a commutative ring with unity. Then
MATH 431 PART 3: IDEALS, FACTOR RINGS, AND RING HOMOMORPHISMS
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The next adjective we can apply to ideals is kind of analogous to cyclic subgroups.
Definition 8. Let R be a commutative ring with unity, and let a ∈ R. Then the ideal {ra | r ∈ R} is the ideal of
all multiples of a, and is called the principal ideal generated by a. We denote this by hai. An ideal I of R is a
principal ideal if I = hai for some a ∈ R.
Remark 3. BE CAREFUL! hai =
6
a, a2 , a3 , . . . ! Even though we are using the same notation as “cyclic group
generated by”, we are generating a subring in a different way than before. Here, hai = {ra | r ∈ R}, the set of things
where you took something from R and multiplied it by a.
Example 7. What is the ideal h2i in Z? What is h1i?
Example 8. What is the ideal hxi in Z[x]?
Instead of having only one “generator” for an ideal, we could have many (like having a finitely generated group).
Theorem 17. Let R be a commutative ring with unity and ai ∈ R for i ∈ {1, 2, . . . , n}. Then
I = ha1 , a2 , . . . , an i = {r1 a1 + r2 a2 + · · · + rn an | ri ∈ R}
is an ideal. (If n is minimal and n > 1, the ideal is not principal.)
You can think of these ideals as being sets of R-linear combinations of a set of basis elements.
Example 9. In Z × Z, the subring {(2a, 0) | a ∈ Z} ∪ {(0, 3b) | b ∈ Z} is an ideal. But this ideal is not principal; in
fact we could write it as h(2, 0), (0, 3)i = {(a1 , b1 )(2, 0) + (a2 , b2 )(0, 3) | ai , bi ∈ Z}.
Example 10. Consider the ideal h2, 3i in Z; what is this ideal? What about h2, 6i?
Definition 9. An integral domain D is a principal ideal domain (PID) if every ideal in D is a principal ideal.
Theorem 18. Z is a PID.
These adjectives are really important and we will be using them a lot. Many of the theorems we will prove will
hinge on using the right kind of ideal, and often the conditions on a theorem will include an ideal being maximal,
prime, or principal.
5. Prime fields
Remember the definition of the characteristic of a ring:
Definition 10. If for a ring R there exists a positive integer n such that n · a = 0 for all a ∈ R, then the least such
positive integer is the characteristic of R. If no such integer exists, then R is of characteristic zero.
Example 11. Z/nZ has characteristic n. Z, Q, and R have characteristic zero.
We seem to keep coming back to Z, Z/nZ, Q, and Z/pZ as our central examples of rings and fields. Why is that?
Theorem 19. If R is a ring with unity and char(R) = n > 1 then R contains a subring isomorphic to Z/nZ. If
char(R) = 0 then R contains a subring isomorphic to Z.
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MATH 431 PART 3: IDEALS, FACTOR RINGS, AND RING HOMOMORPHISMS
Proof sketch. You can show that the map φ : Z → R where φ(n) = n · 1 = 1 + 1 + · · · + 1 is a homomorphism. Then
ker φ is an ideal of Z, and Theorem 10 implies that Z/ ker φ ∼
= φ[Z], a subring of R. If char(R) = n > 1, ker φ = nZ
and if char(R) = 0, ker φ = {0}. Then either Z/nZ or Z/ {0} = Z is isomorphic to a subring of R.
So a “nice” ring has a subring that “feels like” Z or Z/nZ. What about fields?
Corollary 20. The characteristic of a field F is either zero or prime. If char(F ) = p is prime then F contains a
subfield isomorphic to Z/pZ, and if char(F ) = 0 then F contains a subfield isomorphic to Q.
This corollary says that every field contains a subfield isomorphic to either Z/pZ or Q, so these fields are really the
primary building blocks of all the other fields we’re going to study! We call these two fields prime fields to express
this idea.
6. Ideals in polynomial rings
Let F be a field.
Example 12. We described the elements of the principal ideal hxi in Z[x] in Example 8. Does it change if we work
in F [x]?
Example 13. In F [x], describe h2xi. What about x2 ? Or hx − 2i? Does anything change if we work in Z[x]?
Example 14. In Z[x], the subring of polynomials with even constant terms is an ideal. How could we write it in
terms of generators? What if we did this in F [x] for a field F ?
Theorem 21. If F is a field, every ideal in F [x] is principal.
Theorem 22. An ideal hp(x)i =
6 0 of F [x] is maximal if and only if p(x) is irreducible over F .
This theorem (combined with Theorem 13) will be key when we talk about finite fields....
Example 15. You can show that x3 + 3x + 2 is irreducible in Z/5Z[x]. Then Z/5Z[x]/ x3 + 3x + 2 is a field. So is
Q[x]/ x2 − 2 !
These kinds of constructions are our overarching goal. We will spend a lot of time thinking about what kinds of
fields we can construct as factor rings of polynomial rings!