Null ideals of matrices over residue class rings of principal ideal domains

Given a square matrix $A$ with entries in a commutative ring $S$, the ideal of $S[X]$ consisting of polynomials $f$ with $f(A) =0$ is called the null ideal of $A$. Very little is known about null ideals of matrices over general commutative rings. We compute a generating set of the null ideal of a matrix in case $S = D/dD$ is the residue class ring of a principal ideal domain $D$ modulo $d\in D$. We discuss two applications. At first, we compute a decomposition of the $S$-module $S[A]$ into cyclic $S$-modules and explain the strong relationship between this decomposition and the determined generating set of the null ideal of $A$. And finally, we give a rather explicit description of the ring \IntA of all integer-valued polynomials on $A$.


Introduction
Matrices with entries in commutative rings arise in numerous contexts, both in pure and applied mathematics. However, many of the well-known results of classical linear algebra do not hold in this general setting. This is the case even if the underlying ring is a domain (but not a field). For a general introduction to matrix theory over commutative rings we refer to the textbook of Brown [4]. The purpose of this paper is to provide a better understanding of null ideals of square matrices over residue class rings of principal ideal domains. In case S is a field, it is well-known that the null ideal of A is generated by a uniquely determined monic polynomial, the so-called minimal polynomial µ A of A. Further, it is known that if S is a domain, then the null ideal of every square matrix is principal (generated by µ A ) if and only if S is integrally closed, (Brown [5], Frisch [9]). However, little is known about the null ideal of a matrix with entries in a commutative ring. The well-known Cayley-Hamilton Theorem states that every square matrix over a commutative ring satisfies its own characteristic equation (cf. [12,Theorem XIV.3.1]). Therefore there always exists a monic polynomial in S[X] of minimal degree which annihilates the matrix. Note that, in case S is a field, the definition above is consistent with the classical definition of the (uniquely determined) minimal polynomial of a square matrix. However in general, if S is not a field, a minimal polynomial of a matrix over S is not uniquely determined, although its degree is. It is known that if S is a domain, then the null ideal of A is principal if and only if A has a uniquely determined minimal polynomial over S, which is in turn equivalent to the (uniquely determined) minimal polynomial µ A of A over the quotient field of S being in S[X]. Brown discusses conditions for the null ideal to be principal over a general commutative ring R (with identity). In [7], he gives sufficient conditions on certain R[X]-submodules of the null ideal for the null ideal to be principal. There is also earlier work of Brown investigating the relationship of the null ideals of certain pairs of square matrices over a commutative ring (which he refers to as spanning rank partners), see [5], [6]. A better understanding of null ideals of matrices over residue class rings of domains has applications in the theory of integer-valued polynomials on matrix rings. Let D be a domain with quotient field K, and let A ∈ M n (D). The ring Int(M n (D)) and other generalizations of integer-valued polynomial rings are subject of recent research, see [8], [10], [11], [13], [14] and [15]. The connection between integer-valued polynomials on a matrix and null ideals of matrices is the following: Let f ∈ K[X], then there exist g ∈ D[X] and d ∈ D such that f = g/d. The following assertion holds: which is the case if and only if the residue class of g is in the null ideal of A over the residue class ring D /dD. In this paper, we investigate the null ideal of a square matrix A over the residue class ring D /dD of a principal ideal domain D modulo d ∈ D. In Section 2 we provide a description of a specific set of generators of the null ideal of a matrix with entries in D /dD. With this goal in mind, we generalize the notion of the null ideal at the beginning of the section. Instead of looking only at the ideal of polynomials which map A to the zero ideal, we are also interested in those polynomials which map A to the ideal d M n (D), cf. Definition 2.1. This point of view has the advantage that it allows us to work over domains instead of residue class rings (which, in general, have zero-divisors). Further, it turns out that it suffices to consider the special case when d = p ℓ is a prime power (ℓ ∈ N and p ∈ D a prime element). The main result of this section is Theorem 2.19 which describes a specific set of generators of the null ideal of a matrix over D /p ℓ D. However, this description is theoretic; so far, we do not know how to determine them algorithmically in general. It is possible to compute these generators explicitly in case of diagonal matrices. We present this approach at the end of Section 2. The theoretical results in Section 2 allow us to present two applications. In Section 3 we analyze the D /p ℓ D-module structure of D /p ℓ D[A] for A ∈ M n ( D /p ℓ D). As a finitely generated module over a principal ideal ring, D /p ℓ D[A] decomposes into a direct sum of cyclic submodules with uniquely determined invariant factors, according to [4,Theorem 15.33]. We describe this decomposition explicitly and find a strong relationship to the generating set of N D /p ℓ D (A) from Section 2. This allows us to find certain invariant properties of this generating set.
In the last section we apply the knowledge about the null ideal gained in Section 2 to integer-valued polynomials. We give an explicit description of the ring Int(A, M n (D)) using the generating set of the null ideal of A modulo finitely many prime powers p ℓ . Once this description is given, the ring Int-Im(A, M n (D)) of images of A under integer-valued polynomials is easily determined.

Generators of the null ideal
As already mentioned in the introduction, the goal of this section is to compute a generating set of the null ideal of a square matrix over residue class rings of a principal ideal domain D. However, as it is much more convenient to work over domains instead of residue class rings (which, in general, contain zero-divisors) it turns out to be useful to generalize the notion of the null ideal of a matrix. Instead of investigating only ideals of polynomials which map a given matrix to the zero ideal, we are also interested in polynomials which map the matrix to the ideal J M n (D) where J is an ideal of D.
Although the results in this paper are restricted to matrices over principal ideal domains and their residue class rings, the following definitions make sense in much broader generality. Therefore, up to and including Remark 2.6, we allow the underlying ring to be a general commutative ring.
Definition 2.1. Let S be a commutative ring, J an ideal of S and A ∈ M n (S) a square matrix. We call Remark 2.6. The S-ideal N S S (A) of every square matrix A over S is just the whole ring S[X] (that is, if J = (1) = S is the unit ideal). It is therefore generated by the constant polynomial 1. Hence the constant 1 is the (uniquely determined) S-minimal polynomial of every square matrix A over S.
As stated at the beginning of this section, for the remainder of this paper we restrict the underlying ring to be a principal ideal domain. Hence, from this point on, the following notation and conventions hold. The first result of this section is the following lemma. It states a simple but crucial relation between the degrees and the leading coefficients of polynomials in the (d)-ideal of a matrix. Observe that if the leading coefficient of a polynomial g ∈ D[X] (denoted by lc(g)) is coprime to d, then it is a unit modulo d. Hence, there exists an element c ∈ D such that [cg] d is a monic polynomial in D /dD[X]. In particular, this implies the following lemma.
Recall that N 0 (A) = N(A) is the null ideal of A over D. Further, D is integrally closed, since it is a principal ideal domain. As mentioned in the introduction, this implies that the minimal polynomial of every square matrix in M n (D) is in D[X] and generates its null ideal. In particular, holds, where µ A ∈ D[X] is the minimal polynomial of A over K. This completes the case d = 0. For d = 0, we first observe, that it suffices to compute N d (A) for d = p ℓ with p ∈ D a prime element and ℓ ∈ N. Lemma 2.9. Let D be a principal ideal domain, A ∈ M n (D) and a, b ∈ D be coprime elements. Then Proof. The inclusion "⊇" is trivial. For "⊆", let g ∈ N ab (A). Since a and b are coprime, there exist h 1 , h 2 ∈ D[X] such that g = ah 1 + bh 2 . It follows that h 1 ∈ N b (A) and h 2 ∈ N a (A), which completes the proof.
Notation and Conventions 2.10. For the rest of this section we fix the prime element p ∈ D. If A ∈ M n (D) is fixed, we often write N p ℓ instead of N p ℓ (A).
Our goal is to determine polynomials f 0 , . . . , f m ∈ D[X] such that for A ∈ M n (D). Since D /pD is a field, the null ideal of A modulo p is a principal ideal. Hence where ν 1 is a (p)-minimal polynomial of A. The degree of ν 1 is, by definition, independent of the choice of a (p)-minimal polynomial.
Note again, that this definition depends only on the residue class of A modulo p, cf. Remark 2.5. Observe that the following inclusions hold where ν 1 is a (p)-minimal polynomial of A. The p-degree of A is a lower bound for the degree of all polynomials in N p ℓ \ p ℓ D[X], as the following lemma states.
Proof. We prove this by contradiction. Let ℓ ≥ 1 be minimal such that there exists a which is equivalent to h ∈ N p . Then again, by minimality of ℓ > 1, it follows that h ∈ pD[X] and therefore f ∈ p ℓ D[X], contrary to our assumption.
The next proposition provides one of the main tools in this section. It states a simple but important result, which allows us to deduce various properties of the generators of N p ℓ . Proposition 2.13. Let D be a principal ideal domain, p ∈ D a prime element. Further, let A ∈ M n (D) be a square matrix over D, and ν ℓ be a (p ℓ )-minimal polynomial of A (for ℓ ≥ 1). If f ∈ N p ℓ (A), then there exist uniquely determined polynomials q, g ∈ D[X] such that deg(g) < deg(ν ℓ ) and In particular, Proof. Let f ∈ N p ℓ . Since ν ℓ is monic for every ℓ ≥ 1, we can use polynomial division: It is easily seen that r ∈ N p ℓ , hence it suffices to prove the following claim.
If ℓ = 1, then the assertion follows from Lemma 2.12. Let ℓ > 1 be minimal such that the claim is false.
. Therefore, there exists q 1 , q 2 ∈ D[X] with q 2 = 0 and no non-zero coefficient of q 2 is divisible by p such that Hence r can be written in the following form We state a corollary of Proposition 2.13, which is particularly useful: the smaller the degree of a polynomial in N p ℓ , the higher the power of p that divides it.
Corollary 2.14. Let D be a principal ideal domain and p ∈ D a prime element. Further, Observe, that f = pg for some g ∈ N p ℓ−1 , according to Proposition 2.13. Hence if ℓ = j ≥ 1, then the assertion follows. In particular, if ℓ = 1, then j = 1 which proves the basis. Hence assume ℓ > j > 1. Then j ≤ ℓ − 1 and we can apply the induction hypothesis to g ∈ N p ℓ−1 and conclude that g ∈ p ℓ−1−(j−1) D[X] which completes the proof.
At this point, we have enough tools to prove that the polynomials Theorem 2.15 states that the null ideal N p ℓ of A is generated by the ℓ + 1 polynomials p ℓ−i ν i for 0 ≤ i ≤ ℓ. However, in general this is not a minimal generating set. While we are not able to decide which subsets are minimal generating sets, we can still identify some redundant polynomials in This motivates the following definition.
Remark 2.17. The (uniquely determined) degree of a (p j )-minimal polynomial of A depends only on the residue class of A modulo p ℓ , not on the choice of a representative.
Remark 2.18. The indices 0 and ℓ are always contained in I ℓ . Further, the ℓ-th index set I ℓ of A satisfies the following: The ℓ-th index set of A contains the information which (p j )-minimal polynomials we need to generate N p ℓ as stated by the next theorem.
Remark 2.20. For the general case, let d = m i=1 p ℓ i i be the prime factorization of an element d ∈ D and c i = j =i p ℓ j j . Let ν (p,ℓ) denote a (p ℓ )-minimal polynomial and I (p,ℓ) the ℓ-th index set of A with respect to the prime element p. According to Theorem 2.19 and Lemma 2.9, the following holds: The following assertions are technical observations which are useful later-on.
Corollary 2.21. Let D be a principal ideal domain and p ∈ D a prime. Further, let A ∈ M n (D) be a square matrix over D with ℓ-th index set I ℓ (for ℓ ≥ 0) and where I Proof. We prove this by induction on ℓ.
In this case the assertion holds, according to Theorem 2.19. In particular, this is the case if ℓ = 0 (which is the induction basis), since deg(f ) ≥ 0 = deg(ν 0 ). Hence assume ℓ ≥ 1 and deg(f ) < deg(ν ℓ ). Then ℓ / ∈ I [f ] ℓ , and, by Corollary 2.14, f = ph with h ∈ N p ℓ−1 . According to the induction hypothesis, it follows that Note that deg(f ) = deg(h) and therefore I ℓ−1 . We split into two cases, deg(ν ℓ ) > deg(ν ℓ−1 ) and deg(ν ℓ ) = deg(ν ℓ−1 ). According to Remark 2.18 ℓ in this case too. Hence, in both cases, the following holds: For i ≥ 1, let ν i ∈ D[X] be (p i )-minimal polynomials and µ A ∈ D[X] the minimal polynomial of A. Then, by definition, In particular, this sequence of degrees stabilizes. The following proposition states that there always exists an m such that every (p m )-minimal polynomial has degree d A , that is, the sequence stabilizes always at the value d A .
Proposition 2.22. Let D be a principal ideal domain and p ∈ D a prime element.
Since ν m+k+1 − ν m+k ∈ N p m+k is a polynomial with degree less than deg(ν m ), it follows from Corollary 2.14 that are p-adic Cauchy sequences in D. Therefore ν = lim k→∞ ν m+k is a polynomial over the p-adic completion D of D with coefficients a i = lim k→∞ a (k) i and d = deg(ν). Since, ν m+k is a monic polynomial for all k, it follows that ν is a monic polynomial too. Further ν(A) = 0, and hence ν ∈ N D (A). Now, let K be the quotient field of D. Then K is a field extension of K. Since the minimal polynomial is invariant under field extensions, it follows that N K (A) = µ A K[X]. However, D is integrally closed in K, and therefore N D (A) = µ A D[X]. Hence µ A | ν which implies in particular that d A ≤ deg(ν) = d.
We can conclude, that it suffices to determine a finite number of (p i )-minimal polynomials in order to describe the ideals N p ℓ (A) for all ℓ ≥ 0. Corollary 2.23. Let D be a principal ideal domain and p ∈ D a prime element. Further, let A ∈ M n (D) and µ A ∈ D[X] the minimal polynomial of A. Then there exists m ∈ N such that for all k ≥ 0 the following holds: Proof. For i ≥ 0, let ν i be a (p i )-minimal polynomial of A. Then there exists an m ∈ N such that deg(µ A ) = deg(ν m+1 ), according to Proposition 2.22. Hence, µ A is a (p m+k+1 )minimal polynomial for all k ≥ 0 and the assertion follows from Corollary 2.14 (with j = m + 1).

Diagonal matrices
Although we know that (p ℓ )-minimal polynomials exist, it is in general not clear how to determine them algorithmically. However, in the special case of diagonal matrices it is possible to compute them explicitly. Let A = diag(a 1 , . . . , a n ) be a diagonal matrix over D, p ∈ D a prime element, ℓ ∈ N and f ∈ D[X] a polynomial. Then f (A) = diag(f (a 1 ), . . . , f (a n )) holds and therefore However, the set of polynomials which maps the elements a 1 , . . ., a n to multiples of p ℓ can be determined using Bhargava's p-orderings, cf. [1] and [2]. We explain his approach here in the special case of a principal ideal domain (although it is applicable in the more general case of a Dedekind domain by looking at prime ideals instead of prime elements).
Definition 2.24. Let S be a non-empty subset S of D. A p-ordering of S is a sequence (b k ) k≥0 which is defined iteratively in the following way: 1. Choose b 0 ∈ S arbitrary.
2. If b 0 , . . ., b k−1 are already known, then choose b k ∈ S as an element such that In general, there is more than one p-ordering of a set S (except |S| = 1) and for each p- Then (v k (S, p)) k≥0 is called the associated p-sequence of S.
Note that v 0 (S, p) = D. By definition, p-orderings satisfy the following property Therefore, the associated p-sequence of S forms a descending chain of ideals, that is, v k+1 (S, p) ⊆ v k (S, p) for all k ≥ 0. In particular, if S is finite, then v k (S, p) = 0 for k ≥ |S| + 1. Moreover, the property in (2.2) implies that the polynomials of the form In fact, the polynomials f k are indeed a suitable choice for our purpose. The following theorem allows us to deduce the desired properties.
is a polynomials of degree k such that I g = v k (S, p).
We can use this theorem to compute (p ℓ )-minimal polynomials for the diagonal matrix A = diag(a 1 , . . . , a n ) over principal ideal domains. Let S = {a 1 , . . . , a n } be the set of diagonal elements of A and σ a permutation of {1, . . . , n} such that (a σ(i) ) n i=1 is a p-ordering of S. We set f k = (X − a σ(0) )(X − a σ(1) ) · · · (X − a σ(k−1) ). For ℓ ∈ N, let k be minimal such that v k (S, p) ⊆ p ℓ D. Then, by Theorem 2.26, f k (S) ⊆ p ℓ D and we claim that f k is a (p ℓ )-minimal polynomial. Assume that f ∈ D[X] is a monic polynomial with degree less than k and f (S) ⊆ p ℓ D. Again by Theorem 2.26, this implies v k−1 (S, p) ⊆ I f ⊆ p ℓ D which contradicts the choice of k.
To compute the (p ℓ )-minimal polynomial of A we therefore only have to compute a pordering of the set of diagonal elements of A. To demonstrate this approach, we conclude this section with an example of a 3×3-matrix over Z. is the minimal polynomial of A over Q. Since µ A ∈ Z[X], it is the (in this case uniquely determined) minimal polynomial (or 0-minimal polynomial) of A over Z. Let p ∈ Z be a prime element. Recall that we denote the residue classes modulo a prime element p by [ . ] p . Then [A] p has three different eigenvalues in Z /pZ for all prime elements in Z except for the primes 2, 3 and 7. Therefore, is the minimal polynomial of [A] p over Z /pZ for all p ∈ P \ {2, 3, 7}. This implies d p (A) = deg(µ A ) for all p ∈ P \ {2, 3, 7}. Therefore µ A is a (p ℓ )-minimal polynomial of A and {0, ℓ} the ℓ-th index set of A with respect to the prime p for all prime elements p = 2, 3, 7 and all ℓ ≥ 1. Hence, according to Theorem 2.19, holds for all p ∈ P\{2, 3, 7} and all ℓ ≥ 1. The cases p = 3 and p = 7 are similar, therefore, we only handle p = 3. Observe that 4, 32, 16, 16, . . . is an example of a 3-ordering of the set {4, 16, 32} and D, D, (3), 0, 0, . . . is the associated 3-sequence of this set. Following Bhargava's approach (which we explained above this example), it follows that f 2 = (X − 4)(X − 32) is a (3)-minimal polynomial and µ A = f 3 = (X − 4)(X − 32)(X − 16) is a (3 ℓ )-minimal polynomial ℓ ≥ 2. Moreover, {0, 1} is the first and {0, 1, ℓ} is the ℓ-th index set of A for ℓ ≥ 2 (with respect to 3). Theorem 2.19 implies and, for all ℓ ≥ 2, It remains to consider the case p = 2. The sequence 4, 16, 32, 32, . . . is an example of a 2-ordering of the set {4, 16, 32} and D, (4), (64), 0, 0, . . . is the associated 2-sequence of this set. We use Bhargava's approach again; the results are displayed in Table 2.1. Finally, it is worth mentioning that even if the degrees of (p ℓ )-and (p ℓ+1 )-minimal polynomials coincide, a (p ℓ )-minimal polynomials is in general not a (p ℓ+1 )-minimal polynomial (while the reverse implication holds). This is easily verified, once one observes that X 2 is both, an (8)-and a (16)-minimal polynomial, but it is not a (32)-minimal polynomial of A.

Module structure of D /p ℓ D[A]
Throughout this section we fix the prime power p ℓ ∈ D and write R ℓ for the residue class ring D /p ℓ D. Let A ∈ M n (R ℓ ) be a square matrix with null ideal Further, let A ′ ∈ M n (D) be a preimage of A under the projection modulo p ℓ , that is, where I ℓ is the ℓ-th index set of A ′ and ν i are (p i )-minimal polynomials of A ′ (for i ∈ I ℓ \ {0}).
Notation and Conventions 3.1. Let f ′ ∈ D[X] be a monic polynomial. Recall that, For a better readability, we often write p for the residue class [p] p ℓ of p modulo p ℓ and say Note that the ℓ-th index set of a matrix A ′ ∈ M n (D) only depends on the residue , then A ′ and A ′′ have equal ℓ-th index sets, cf. Remark 2.17.
. In this section we analyze the structure of the R ℓ -module R ℓ [A]. Since the null ideal of A contains a monic polynomial, there exists a power of A which can be written as an R ℓ -linear combination of smaller powers of A. Therefore the module R ℓ [A] is finitely generated. As a finitely generated module over a principal ideal ring, R ℓ [A] decomposes into cyclic R ℓ -submodules, according to [4,Theorem 15.33]. We compute such a decomposition exploiting its relation to the generating set of the null ideal N of A which we determined in Theorem 2.19 of the last section. In particular, it turns out that the invariant factors of R ℓ [A] correspond to the elements in the reduced index set I ⋆ ℓ of A. Further, their multiplicities relate to the degrees of the (p j )-minimal polynomials, see Remark 3.6. As the invariant factors are uniquely determined, this corroborates the usefulness of the set of generators of the null ideal of A which we determined in Section 2.
To be more specific, Theorem 3.5 below states that, if I ⋆ ℓ is the reduced index set of A and s j = deg(ν succ(j) ) − deg(ν j ) for j ∈ I ⋆ ℓ , then where d p = deg(ν 1 ) is the degree of the minimal polynomial of A modulo p. Roughly speaking, the R ℓ -free part R dp ℓ of the decomposition in (3.1) indicates what happens in terms of classical linear algebra over the field R 1 while the torsion-part of R ℓ [A] relates to the set I ⋆ ℓ . In order to understand this connection, let d be the degree of a (p ℓ )-minimal polynomial ν ℓ . Then A d is an R ℓ -linear combination of I, A, ..., A d−1 , and thus R ℓ Hence the following sequence of R ℓ -modules is exact.
where e 1 , . . ., e d is an arbitrary basis of R d ℓ . It follows that Elements of ker(ψ) correspond to relations between the matrices I, A, . . ., A d−1 and therefore to polynomials in the null ideal N of A of degree less than d. Hence where λ 1 , . . . , λ d ∈ R ℓ . We exploit this equivalence and use a generating set of the null ideal N of A to compute a generating set of the module ker(ψ). Nevertheless, we need to be careful, since (as an ideal of R ℓ [X]) N is an R ℓ [X]-module and ker(ψ) is only an R ℓ -module. Hence multiplication by X needs to be dealt with when transferring a generating set of N to a generating set of ker(ψ). For this purpose, set is an R ℓ -module isomorphism. Let be the set of all elements in N of degree less than d. Then N <d is an R ℓ -module, and for f 1 , . . . , f r ∈ R ℓ [X] <d , the following holds . . , f r R ℓ ⇐⇒ ker(ψ) = ϕ(f 1 ), . . . , ϕ(f r ) R ℓ according to the equivalence in (3.3). We modify the sequence in (3.2) accordingly to get the following exact sequence of R ℓ -modules.
The following lemma describes which R ℓ [X]-generating sets of N can be transferred to R ℓ -generating sets of N <d . Then Proof. The conditions on the degrees of the polynomials f i guarantee that deg(X t−1 f i ) < d for 1 ≤ i ≤ m − 1 and 1 ≤ t ≤ s i . Hence the inclusion "⊇" is easily seen and it suffices to show "⊆". Let f ∈ N <d . We prove this by induction on deg(f ).
For the basis, let 0 = f ∈ N <d be a polynomial of minimal degree in N <d , that is, it follows that and according to our assumptions on the polynomials f i (where we write p for its residue class (3.6) Therefore f = qf k + p t k r which implies p t k r ∈ N <d , and we can apply the induction hypothesis to p t k r. Hence Since and the assertion follows for f = qf k + p t k r.
According to Corollary 2.21, any generating set of the form { p ℓ−i ν i | i ∈ I ⋆ ℓ }, where ν i ∈ R ℓ [X] are (p i )-minimal polynomials, satisfies the conditions of Lemma 3.4. This allows us to prove the following theorem which is the main result of this section.
Further, let I ⋆ ℓ be the reduced index set of A and s i = deg(ν succ(i) ) − deg(ν i ) for i ∈ I ⋆ ℓ , then Proof. First, we show that the two decompositions of R ℓ [A] given in the theorem, are isomorphic. Recall that ν 0 = 1 and d 0 = 0. Hence R dp , and if one of these equivalent conditions is satisfied, then d i+1 = d succ(i) .
Hence the two representations are isomorphic and it suffices to show that According to Corollary 2.21 the polynomials in { p ℓ−i ν i | i ∈ I ⋆ ℓ } satisfy the conditions of Lemma 3.4, and therefore Observe that deg(b j ) = j − 1. Hence b 1 , . . . , b d is a basis of R ℓ [X] <d . Together with the exact sequence (3.5), this implies Remark 3.6. Let the notation be as in Theorem 3.5.
Recall that the ℓ-th index set of a matrix defines a generating set of the null ideal N R ℓ (A) of A consisting of polynomials of the form p ℓ−j ν j . Per definition, I ⋆ ℓ depends on the degrees of these polynomials. In particular, observe that I ⋆ ℓ = ∅ if and only if deg(ν ℓ ) = deg(ν 1 ) = d p . Together with Theorems 2.19 and 3.5 this implies the following corollary.
Corollary 3.7. Let A ∈ M n (R ℓ ) with ℓ-th index set I ℓ , (p ℓ )-minimal polynomial ν ℓ and p-degree d p . Then the following assertions are equivalent: We can reformulate this in terms of matrices with entries in D.
Corollary 3.8. Let A ∈ M n (D) and ℓ ∈ N. Further, let ν j ∈ D[X] be (p j )-minimal polynomials of A for 1 ≤ j ≤ ℓ and [A] p j be the image of A under projection modulo p j . The following assertions are equivalent.
Recall, that Proposition 2.22 states, that for A ∈ M n (D), there exists m ∈ N such that deg(ν m+k ) = deg(ν A ) for all k ≥ 0. Then I ⋆ m+k = I ⋆ m , cf. Remark 2.18. Together with Theorem 3.5 we conclude this section with a final corollary. Corollary 3.9. Let A ∈ M n (D) and ν j be (p j )-minimal polynomials for j ≥ 1. Further, let [A] p j be the image of A under projection modulo p j . Then there exists m ∈ N such that for all ℓ ≥ m the following holds where I ⋆ m is the reduced index set of [A] p m and s j = deg(ν succ(j) ) − deg(ν j ) for j ∈ I ⋆ m . In particular, R ℓ [[A] p ℓ ] decomposes into deg(µ A ) non-zero cyclic summands.

Integer-valued polynomials on one matrix
This section is dedicated to the application of the results of Section 2 in the context of integer-valued polynomials on a single matrix. Again, let D be a principal ideal domain with quotient field K and A ∈ M n (D) be a square matrix with entries in D. We want to determine the ring Int(A, M n (D)) of all integer-valued polynomials on A, that is, There are both instances in which equality holds, and instances in which the inclusion is strict. If equality holds, it is readily seen that Int-Im(A, M n (D)) = D[A], that is, all images of A under integer-valued polynomials on A can be written as g(A) with g ∈ D[X]. As far as the images of A are concerned, the integer-valued polynomials in K[X] \ D[X] do not contribute anything new in this case. In fact, as the next proposition states, the reverse implication holds too. (Thanks to Giulio Peruginelli for pointing this out.) holds, and we can restrict the inner sum in Equation (4.3) to all 1 ≤ ℓ ≤ m p . And finally, since pN p ℓ−1 (A) ⊆ N p ℓ (A), it follows hat 1 p ℓ−1 N p ℓ−1 (A) ⊆ 1 p ℓ N p ℓ (A). Hence  where ν (p,j) ∈ D[X] are (p j )-minimal polynomial of A for j ≥ 0, and I (p,mp) is the m p -th index set of A with respect to the prime p.