Comm. Algebra 39(4) 2011, pp 1482–1490

Non-unique factorization of polynomials over residue class rings of the integers

Christopher Frei and Sophie Frisch

Abstract. We investigate non-unique factorization of polynomials in ℤ_{p^{n}}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, ℤ_{p^{n}}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of ℤ_{p^{n}}[x] is a direct sum of monoids corresponding to irreducible polynomials in ℤ_{p}[x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ ℕ, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.

Keywords: polynomial, finite ring, non-unique factorization, elasticity, zerodivisor, monoid. MSC 2000: Primary 13M10; Secondary 13P05, 11T06, 20M14.

1. Introduction

Polynomials with coefficients in ℤ_{p^{n}} = ℤ/p^{n}ℤ, p prime, n ≥ 2, have been investigated quite a lot with respect to the polynomial functions they define [6, 9, 4]. Comparatively little is known about the multiplicative monoid of non-zerodivisors of ℤ_{p^{n}}[x]. McDonald [7] and Wan [8] list a few properties concerning uniqueness of certain kinds of factorizations of polynomials in ℤ_{p^{n}}[x]. At the same time, factorization into irreducibles is manifestly non-unique. In this paper, we investigate non-uniqueness of factorization of polynomials in ℤ_{p^{n}}[x] into irreducibles, in the spirit of [5]. Throughout this paper, p is a fixed prime and n ≥ 2.

Here is an example of the phenomena we will study. Consider the identity in ℤ_{p^{n}}[x]:

(x^{m} + p^{n−1})(x^{m} + p) = x^{m}(x^{m} + p^{n−1} + p).

NON-UNIQUE FACTORIZATION OF POLYNOMIALS

Assuming for the moment that every element of ℤ_{p^{n}}[x] has a factorization into irreducible elements (cf. Corollary 2.9), we note that (x^{m} + p^{n−1}) is a product of at most n−1 irreducibles, and that (x^{m}+p) is irreducible. This is so because both polynomials represent a power of x in ℤ_{p}[x]. By unique factorization in ℤ_{p}[x], each of their factors in ℤ_{p^{n}}[x] (apart from units) also represents a power of x in ℤ_{p}[x]. (We have used the fact that a polynomial in ℤ_{p^{n}}[x] is a unit if and only if it maps to a unit in ℤ_{p}[x] under canonical projection.) Since every non-unit factor in ℤ_{p^{n}}[x] of (x^{m} + p^{n−1}) or (x^{m} + p) represents a power of x in ℤ_{p}[x], the constant coefficient of every such factor is divisible by p. On the other hand, the constant coefficient of (x^{m} + p^{n−1}) is divisible by no higher power of p than p^{n−1}, so it cannot be a product of more than n − 1 non-unit polynomials, and (x^{m} + p) cannot be a product of more than one non-unit in ℤ_{p^{n}}[x]. We have seen that for arbitrary m ∈ ℕ, there exists in ℤ_{p^{n}}[x] a product of at most n irreducibles that is also representable as a product of more than m irreducibles.

We now establish some notation:

1.1. Definition. We denote p-adic valuation on ℚ by v and extend v to ℚ(x) by defining v(f) = min_{k} v(a_{k}) for f = ∑_{k} a_{k}x^{k} ∈ ℤ[x], and v(f/g) = v(f) − v(g). This gives a surjective mapping v: ℤ[x] → ℕ_{0} ∪ {∞}.

We denote by (N_{n}, +, ≤) the ordered monoid with elements 0, 1, …, n − 1, ∞ resulting from factoring (ℕ_{0}∪{∞}, +, ≤) by the congruence relation that identifies all values greater or equal n, including ∞. By abuse of notation, we still use v for the surjective mapping v: ℤ_{p^{n}}[x] → N_{n} obtained by factoring p-adic valuation v: ℤ[x] → ℕ_{0} ∪ {∞} by the above congruence relation.

The mapping v: ℤ_{p^{n}}[x] → N_{n} behaves like a valuation, except that (N_{n}, +, ≤) is not a group (nor can it be extended to a group, as it is not cancellative).

1.2. Remark. v: ℤ_{p^{n}}[x] → N_{n} satisfies (i) v(f) = ∞ ⟺ f = 0 (ii) v(f + g) ≥ min(v(f), v(g)) (iii) v(f · g) = v(f) + v(g)

We will write p for a prime in ℤ as well as for its residue class in ℤ_{p^{n}} or ℤ_{p^{n}}[x]. This should not cause any confusion; we are always going to specify which ring we are talking about. In the remainder of this section, we review a few well-known facts about ℤ_{p^{n}}[x].

1.3. Fact. For f ∈ ℤ_{p^{n}}[x], the following are equivalent:

(a) v(f) > 0, i.e., all coefficients of f are divisible (in ℤ_{p^{n}}) by p.

(b) f is nilpotent.

(c) f is a zero-divisor.

Proof. This follows from the properties of v: ℤ_{p^{n}}[x] → N_{n} mentioned in 1.2. □

1.4. Definition. Let R be a commutative ring.

(i) Nil(R) denotes the nilradical of R, i.e., the intersection of all prime ideals of R.

(ii) J(R) denotes the Jacobson radical of R, i.e., the intersection of all maximal

ideals of R. (iii) Z(R) denotes the set of zero-divisors of R. (iv) U(R) denotes the group of units of R.

As we all know, the intersection of all prime ideals of a commutative ring coincides with the set of nilpotent elements:

Nil(R) = ⋂_{P prime} P = {r ∈ R | ∃n ∈ ℕ r^{n} = 0}.

1.5. Fact.

Nil(ℤ_{p^{n}}[x]) = J(ℤ_{p^{n}}[x]) = (p) = Z(ℤ_{p^{n}}[x])

Proof. Since the maximal ideals of ℤ_{p^{n}}[x] are precisely the ideals (p, f) with f representing an irreducible polynomial in ℤ_{p}[x], we see J(ℤ_{p^{n}}[x])=(p). We already know (p) = Nil(ℤ_{p^{n}}[x]) = Z(ℤ_{p^{n}}[x]) from Fact 1.3. □

1.6. Fact. Let R be a commutative ring. The units of R[x] are precisely the polynomials a_{0} + a_{1}x + … + a_{n}x^{n} with a_{0} a unit of R and a_{k} nilpotent for k > 0.

Proof. If f is a unit then its image under projection to (R/P)[x] is a unit for every prime ideal P of R. Therefore, a_{0} is not in any P and hence a unit; and for k > 0, a_{k} is in every P and therefore nilpotent. Conversely, if f = a_{0} + g with a_{0} a unit of R and all coefficients of g in the intersection of all prime ideals of R, then g is in every prime ideal of R[x] and hence f = a_{0} + g is in no prime ideal of R[x] and therefore a unit of R[x]. □

1.7. Corollary. Let p be a prime and n ≥ 1. Then U(ℤ_{p^{n}}[x]) consists of precisely the polynomials f = a_{0} + a_{1}x + … + a_{n}x^{n} such that (in ℤ_{p^{n}}) p | a̸_{0} and p | a_{k} for k > 0. It follows that a polynomial in ℤ[x] represents a unit in ℤ_{p^{n}}[x] for some n ≥ 1 if and only if it represents a unit in ℤ_{p^{n}}[x] for all n.

2. Different concepts of irreducibility in rings with zero-divisors

Various facts about irreducible and prime elements that are clear in integral domains tend to become messy and obfuscated as soon as zero-divisors are involved. We will see, however, that everything works just as smoothly as in integral domains if all zero-divisors are contained in the Jacobson radical. For an in-depth investigation of factorization in rings with zero-divisors see [2, 3, 1]. The very natural concepts defined in this section appear in the literature ([2, 3, 1] and their references) under varying names. Our choice of names has been motivated by a desire to define associated and irreducible for commutative rings just like they are usually defined for integral domains and to reflect the existence of a group of “stronger” concepts (associated, irreducible, atomic), and a parallel group of “weaker” concepts, with analogous relationships between the concepts inside each group. R is always a commutative ring with identity.

2.1. Definition. If c ∈ R is a non-zero non-unit then we say that c is weakly irreducible if for all a, b ∈ R

c = ab ⟹ c | a or c | b

and that c is irreducible if for all a, b ∈ R

c = ab ⟹ b is a unit or a is a unit.

A non-zero non-unit c is called prime if for all a, b ∈ R

c | ab ⟹ c | a or c | b

2.2. Definition. Let a, b ∈ R. We call a and b weakly associated if a | b and b | a (or equivalently, if (a) = (b)). We call a and b associated if there exists a unit u ∈ R such that a = bu.

It is clear that irreducible implies weakly irreducible, that associated implies weakly associated, and that both converses hold in integral domains. Also, prime implies irreducible in integral domains. We now consider a class of commutative rings properly containing the class of integral domains in which the implications mentioned above still hold.

2.3. Definition. Let R be a commutative ring. We say that R is a ring with harmless zero-divisors, if Z(R) ⊆ 1 − U(R) = {1 − u | u a unit of R}. (These are the rings called présimplifiable by Bouvier (cf. [2]).)

2.4. Lemma. ([2]). Let R be a ring with harmless zero-divisors and a, b, c, u, v ∈R. Then

(1) if a ≠ 0, a = bu and b = av then u, v are units;

(2) a, b are weakly associated if and only if they are associated;

(3) c is weakly irreducible if and only if c is irreducible;

(4) if c is prime then c is irreducible.

Proof. (1) a = avu; therefore a(1 − vu) = 0, which makes (1 − vu) a zero-divisor. Now 1 − vu = 1 − w for some unit w; hence vu = w and u, v are units. (2), (3) and (4) follow from (1). □

2.5. Corollary. If a commutative ring R satisfies Z(R) ⊆ J(R) then the statements of Lemma 2.4 hold for R. In particular, they hold for ℤ_{p^{n}}[x].

Proof. It is easy to see that J(R) ⊆ 1 − U(R) holds in every commutative ring. Therefore every ring satisfying Z(R)⊆J(R) is a ring with harmless zero-divisors. In particular, ℤ_{p^{n}}[x] is a ring with harmless zero-divisors, since Z(ℤ_{p^{n}}[x]) = J(ℤ_{p^{n}}[x]) by Fact 1.5. □

2.6. Definition. Let R be a commutative ring. R is called atomic if every non-zero non-unit is a product of irreducible elements. We call R weakly atomic if every non-zero non-unit is a product of weakly irreducible elements.

If one examines the usual proof that an integral domain with ACC for principal ideals is atomic, one finds that it shows that any commutative ring with ACC for principal ideals is weakly atomic. Namely, suppose there exists an element s in the set S of those non-zero non-units of R that do not factor into weakly irreducible elements. Then s is not weakly irreducible, so there exist a, b with s = ab and s | a̸, s | b̸. Such a, b must be non-zero non-units, and at least one of them is in S; say a ∈ S. As a | s and s | a̸, we have (s) ⊂ (a). By iteration we get an infinite ascending chain of principal ideals. We have shown:

2.7. Lemma. ([2] Thm. 3.2). If R is a commutative ring satisfying ACC for principal ideals then R is weakly atomic.

Combining Lemmas 2.4 and 2.7, we obtain

2.8. Lemma. ([2]). Every commutative ring with harmless zero-divisors satisfying ACC for principal ideals is atomic; in particular, every Noetherian ring with harmless zero-divisors is atomic.

In particular, Lemma 2.8 applies to ℤ_{p^{n}}[x]:

2.9. Corollary. ℤ_{p^{n}}[x] is atomic.

3. Uniqueness of factorization

In the previous section we have seen that every non-zero non-unit of ℤ_{p^{n}}[x] is a product of irreducibles. Step by step we now reduce the task of factoring arbitrary elements of ℤ_{p^{n}}[x]: first to factoring non-zerodivisors, then to factoring monic polynomials and, finally, to factoring monic primary polynomials. This is a matter of applying well-known facts of the “uniqueness” kind from [7] or [8]. It is only in the next section, concerned with factoring monic primary polynomials, that non-uniqueness of factorization comes into play.

Arbitrary polynomials to non-zerodivisors.

3.1. Lemma. Let p be a prime and n ≥ 2. For f ∈ ℤ_{p^{n}}[x], the following are equivalent:

(1) f = pu for some unit u of ℤ_{p^{n}}[x].

(2) f is prime.

(3) f is irreducible and a zero-divisor.

Proof. (1 ⇒ 2). Let v: ℤ_{p^{n}}[x] → N_{n} be p-adic valuation as defined in 1.1. Since v(p) = 1, additivity of v (1.2) implies that p is prime. Therefore every element associated to p is prime.

(2 ⇒ 3). By Corollary 2.5, prime elements of ℤ_{p^{n}}[x] are irreducible. If c is prime, then (c) ⊇ Nil(ℤ_{p^{n}}[x]) = (p); so c | p and therefore c and p are associated. As p is a zero-divisor, so is c.

(3 ⇒ 1). If c is a zero-divisor, then c ∈ Z(ℤ_{p^{n}}[x]) = (p), meaning c = pv for some v. If, furthermore, c is irreducible, then v must be a unit. □

3.2. Fact. Let f ∈ ℤ_{p^{n}}[x], f ≠ 0.

(i) There exists a non-zerodivisor g ∈ ℤ_{p^{n}}[x], and 0 ≤ k < n, such that f = p^{k}g. Furthermore, k is uniquely determined by k = v(f), and g is unique modulo (p^{n−k}).

(ii) In every factorization of f into irreducibles, exactly v(f) of the irreducible factors are associates of p.

Proof. (i) follows from Fact 1.3 and the definition of p-adic valuation. (ii) follows from (i) and the fact that p is prime in ℤ_{p^{n}}[x] (by Lemma 3.1). □

Nonzerodivisors to monic polynomials.

3.3. Remark. ([7], Thm. XIII.6). Every non-zerodivisor f ∈ ℤ_{p^{n}}[x] is uniquely representable as f = ug, with u a unit of ℤ_{p^{n}}[x] and g monic. The degree of g is deg f̄, where f̄ is the image of f in ℤ_{p}[x] under canonical projection.

Actually, McDonald [7] only shows the existence of g and u in 3.3, but uniqueness is easy: suppose ug = vh with u, v units, g, h monic. Then v^{−1}ug = h. As g, h are monic, so is v^{−1}u. By 1.7, the only monic unit of ℤ_{p^{n}}[x] is 1, however, so u = v and g = h.

Applying 3.3 to the irreducible factors of a non-zerodivisor of ℤ_{p^{n}}[x], we obtain:

3.4. Proposition. Let f ∈ ℤ_{p^{n}}[x], not a zero-divisor, and u and g the unique unit and monic polynomial, respectively, in ℤ_{p^{n}}[x] with f = ug. For every factorization f = c_{1} · . . . · c_{k} into irreducibles, there exist uniquely determined monic irreducible d_{1}, …, d_{k} ∈ ℤ_{p^{n}}[x] and units v_{1}, …, v_{k} in ℤ_{p^{n}}[x] with c_{i} = v_{i}d_{i}, u = v_{1} · . . . · v_{k} and g = d_{1} · . . . · d_{k}.

By Proposition 3.4, we have reduced the question of factoring non-zerodivisors of ℤ_{p^{n}}[x] into irreducibles to the question of factoring monic polynomials into monic irreducibles. As there are only finitely many monic polynomials of each degree in ℤ_{p^{n}}[x], we also see that every non-zerodivisor of ℤ_{p^{n}}[x] has (up to associates) only finitely many factorizations into irreducibles.

Monic polynomials to primary monic polynomials.

3.5. Lemma. Let f ∈ ℤ_{p^{n}}[x], not a zero-divisor. Then (f) is a primary ideal of ℤ_{p^{n}}[x] if and only if the image of f under the canonical projection π: ℤ_{p^{n}}[x]→ℤ_{p}[x] is associated to a power of an irreducible polynomial.

Proof. In the principal ideal domain ℤ_{p}[x], the non-trivial primary ideals are precisely the principal ideals generated by powers of irreducible elements. The canonical projection π induces a bijective correspondence between primary ideals of ℤ_{p}[x] and primary ideals of ℤ_{p^{n}}[x] containing (p).

Now an ideal of ℤ_{p^{n}}[x] that does not consist only of zero-divisors is primary if and only if its radical is a maximal ideal (since the only non-maximal prime ideal is (p) = Z(ℤ_{p^{n}}[x])). Let f ∈ ℤ_{p^{n}}[x]. Then, since every prime ideal of ℤ_{p^{n}}[x] contains (p), the radical of (f) is equal to the radical of (f) + (p) = π^{−1}(π((f))).

For a non-zerodivisor f, therefore, (f) is primary if and only if (f) + (p) is primary, which is equivalent to π(f) being a primary element of ℤ_{p}[x]. □

3.6. Definition. We call a non-zerodivisor of ℤ_{p^{n}}[x] primary if its image under projection to ℤ_{p}[x] is associated to a power of an irreducible polynomial.

The simplest form of Hensel’s Lemma shows that every monic f ∈ ℤ_{p^{n}}[x] is a product of primary polynomials. Furthermore, the monic primary factors of a monic polynomial in ℤ_{p^{n}}[x] are uniquely determined.

3.7. Remark. ([8], Thm. 13.8). Let f ∈ ℤ_{p^{n}}[x] monic, then there exist monic polynomials f_{1}, …, f_{r} ∈ ℤ_{p^{n}}[x] such that f = f_{1} · . . . · f_{r} and the residue class of f_{i} in ℤ_{p}[x] is a power of a monic irreducible polynomial g_{i} ∈ ℤ_{p}[x], with g_{1}, …, g_{r} distinct. The polynomials f_{1}, …, f_{r} ∈ ℤ_{p^{n}}[x] are (up to ordering) uniquely determined.

3.8. Proposition. Every non-zero polynomial f ∈ ℤ_{p^{n}}[x] is representable as

f = p^{k}uf_{1} … f_{r}

with 0 ≤ k < n, u a unit of ℤ_{p^{n}}[x], r ≥ 0, and f_{1}, …, f_{r} ∈ ℤ_{p^{n}}[x] monic polynomials such that the residue class of f_{i} in ℤ_{p}[x] is a power of a monic irreducible polynomial g_{i} ∈ ℤ_{p}[x], and g_{1}, …, g_{r} are distinct.

Moreover, k ∈ ℕ_{0} is unique, u ∈ ℤ_{p^{n}}[x] is unique modulo p^{n−k}ℤ_{p^{n}}[x], and also the f_{i} are unique (up to ordering) modulo p^{n−k}ℤ_{p^{n}}[x].

Proof. Follows from 3.2, 3.4 and 3.7. □

For the the monoid of non-zerodivisors of ℤ_{p^{n}}[x] this means:

3.9. Theorem. Let M be the multiplicative monoid of non-zerodivisors of ℤ_{p^{n}}[x], and U its group of units. Let M′ be the submonoid of M consisting of all monic polynomials. Then

M ≃ U ⊕ M′.

For every monic irreducible polynomial f ∈ ℤ_{p}[x] let M_{f} be the submonoid of M′ consisting of those monic polynomials g ∈ ℤ_{p^{n}}[x] whose image under projection to ℤ_{p}[x] is a power of f. Then

M′ ≃ ∑_{f} M_{f}

where f ranges through all monic irreducible polynomials of ℤ_{p}[x].

Proof. Follows by specializing 3.8 to non-zerodivisors, which by 1.3 are exactly the elements of ℤ_{p^{n}}[x] not divisible by p. □

The example in the introduction shows that the monoid M_{x} (corresponding to the irreducible polynomial x ∈ ℤ_{p}[x]) has infinite elasticity (defined in 4.1 below). We will show this of each of the direct summands M_{f} in Theorem 3.9.

4. Non-uniqueness of factorization and infinite elasticity

Irreducible elements are defined in an arbitrary monoid in the same way as we defined them for the multiplicative monoid of a ring in 2.1. Note that the monoid of non-zerodivisors of ℤ_{p^{n}}[x], and its submonoid the monoid of all monic polynomials in ℤ_{p^{n}}[x] are cancellative.

4.1. Definition. Let (M, ·) be a cancellative monoid.

(1) For k ≥ 2, let ρ_{k}(M) be the supremum of all those m ∈ ℕ for which there exists a product of k irreducibles that can also be expressed as a product of m irreducibles.

(2) The elasticity of M is sup_{k≥2}(ρ_{k}(M)/k); in other words, the elasticity is the supremum of the values m/k such that there exists an element of M that can be expressed both as a product of k irreducibles and as a product of m irreducibles.

4.2. Lemma. Let f be a monic polynomial in ℤ[x] which represents an irreducible polynomial in ℤ_{p}[x]. Let d = deg(f). Let n, k ∈ ℕ with 0 < k < n, m ∈ ℕ with gcd(m, kd) = 1, and c ∈ ℤ with p | c̸. Then

f(x)^{m} + cp^{k}

represents an irreducible polynomial in ℤ_{p^{n}}[x].

Proof. Suppose otherwise. Then there exist g, h, r ∈ ℤ[x], with g, h monic, g irreducible in ℤ_{p^{n}}[x], such that

f(x)^{m} + cp^{k} = g(x)h(x) + p^{n}r(x)

and 0 < deg g < dm. In ℤ_{p}[x], g is a power of f, by unique factorization. Therefore deg g = ds with 0 < s < m.

Let α be a zero of g. Let R be the ring of algebraic integers in ℚ[α] and P a prime of R above p with ramification index e. Let v be the normalized valuation on ℚ[α] corresponding to P.

Since f(α)^{m} = p^{n}r(α) − cp^{k}, we have v(f(α)) = ke/m. As m is relatively prime to k, m divides e. This holds for the ramification index e_{i} of every prime P_{i} of R above p. Let f_{i} be the inertia degree of P_{i}. Then m divides ∑_{i} e_{i}f_{i} = [ℚ[α] : ℚ] = deg g = ds. As m is relatively prime to d, m divides s, a contradiction to 0 < s < m. □

4.3. Theorem. Let n ≥ 2. Let f be a monic irreducible polynomial in ℤ_{p}[x]. Let M_{f} be the submonoid of the multiplicative monoid of ℤ_{p^{n}}[x] defined in Theorem 3.9.

Then the elasticity of M_{f} is infinite. Moreover, ρ_{2}(M_{f}) = ∞.

Proof. In the proof, f will, by abuse of notation, denote a monic polynomial in ℤ[x] which under canonical projection to ℤ_{p}[x] maps to the f in the theorem.

Let q be a prime with q > max(n − 1, deg f). By Lemma 4.2, f(x)^{q} + p^{n−1} is irreducible in ℤ_{p^{n}}[x], and

(f(x)^{q} + p^{n−1})^{2} = f(x)^{q}(f(x)^{q} + 2p^{n−1})

is an example of factorization of a polynomial in M_{f} into either 2 irreducible factors - on the left - or more than q irreducible factors - on the right (actually, the number of irreducible factors on the right is q + 1 for p ≠ 2, by Lemma 4.2, and 2q for p = 2.) As q can be made arbitrarily large, ρ_{2}(M_{f}) = ∞, and the elasticity of M_{f} is infinite. □

4.4. Corollary. Let M be the multiplicative monoid of non-zerodivisors of ℤ_{p^{n}}[x] and M′ the submoinoid of monic polynomials. Then the elasticity of M′ is infinite, and ρ_{2}(M′) = ∞. Therefore the elasticity of M is infinite and also ρ_{2}(M) = ∞.

Proof. Indeed, by Theorem 3.9 and Theorem 4.3, M′ is an infinite direct sum of monoids M_{f} of infinite elasticity and satisfying ρ_{2}(M_{f}) = ∞. □

Acknowledgment. The authors wish to thank F. Halter-Koch for useful comments.

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