On classiﬁcation of ﬁnite commutative chain rings

: Let R be a ﬁnite commutative chain ring with invariants p , n , r , k , m . It is known that R is an extension over a Galois ring GR ( p n , r ) by an Eisenstein polynomial of some degree k . If p (cid:45) k , the enumeration of such rings is known. However, when p | k , relatively little is known about the classiﬁcation of these rings. The main purpose of this article is to investigate the classiﬁcation of all ﬁnite commutative chain rings with given invariants p , n , r , k , m up to isomorphism when p | k . Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of ﬁnite (complete) chain rings with ( p − 1) (cid:45) k is determined. In addition, we study the case ( p − 1) | k , and show that the classiﬁcation is strongly dependent on Eisenstein polynomials not only on p , n , r , k , m . In this case, we classify ﬁnite (incomplete) chain rings under some conditions concerning the Eisenstein polynomials. These results yield immediate corollaries for p-adic ﬁelds, coding theory and geometry.


Introduction
We consider only commutative rings which have an identity. An associative Artinian ring with an identity whose lattice of ideals forms a unique chain is called a chain ring. It is not hard to show that a finite ring R is a chain ring if and only if its (Jacobson) radical J(R) is principal and the quotient F = R/J(R) is a field of order p r , p is prime, i.e., R is a local ring. There are known positive integers p, n, r, k, m associted with R called the invariants of R. Ayoub [3] dubbed these rings "homogenous rings". Finite chain rings appear in various areas, for details see [4]. In particular, finite chain rings have been increasingly used in coding theory [5,11,14] and in geometry as coordinatizing rings of Pappian Hjelmslev planes [9]. An interesting class of finite chain rings is that contains Galois rings, i.e., GR(p n , r) = Z p n [x]/( f (x)), (1.1) where f (x) is a monic polynomial of degree r and irreducible modulo p. However, there are different algebraic ways to construct finite chain rings. Let R denote a finite chain ring with invariants p, n, r, k and m, then R is an Eisenstein extension of a Galois ring GR(p n , r) R = GR(p n , r)[x]/(g(x), x m ), (1.2) where g(x) is an Eisenstein polynomial of degree k over GR(p n , r), i.e., s i x i , where s 0 is a unit of GR(p n , r).
However, the case when g(x) = x k − ps 0 , R is called a pure chain ring. Another construction of R is that connected to p-adic fields; R is a factor-ring of the ring of integers of a suitable finite extension of Q p , the field of p-adic numbers. Let K be an extension of Q p with ramification index k and residue degree r, and let L be the unramified subextension of degree r over Q p . Let O K denote the ring of integers of K, and let π be a prime element of O K . Then, (1.4) where g(x) is the minimal polynomial of π over L whose image g( is Eisenstein of degree k. We shall see later that when m is sufficiently large, the classification of Q pisomorphism finite extensions of Q p coincides with that of finite commutative chain rings. For the basic of p-adic fields, we refer to [8,10]. Let U(R) be the group of units of R, then U(R) =< a > ⊗H, (1.5) where a is an element of order p r − 1, and H = 1 + J(R) (Ayoub [3]). The structure of H is introduced in [3] when (p − 1) k, and given in [1] if (p − 1) | k. However, it turned out that if p | k, H plays a paramount role in the enumeration of finite chain rings. Let k = k 1 p l , (p, k 1 ) = 1, Hou [7] classified pure chain rings when l = 1 and (p − 1) k. Moreover, if l = 0, the classification is independent of H, and the enumeration, in this case, was determined by Clark and Liang [4]. Our main aim in this article is to classify, in case of p | k, finite chain rings with fixed invariants p, n, r, k, m up to ismorphism by using ideas from [1,3]. The present manuscript is organized as follows. Section 2 involves notations and known statements that will appear in the sequel. In Section 3, we consider the classification problem of finite chain rings with the same invariants p, n, r, k, m. First, we classify them when (p − 1) k. Next, the case (p − 1) | k is investigated under certain conditions. Section 4 is devoted to apply our results in p-adic number fields.

Preliminaries
This section collects some facts and states notations required in our subsequent discussions. Let R be a finite chain ring with invariants p, n, r, k and m, and nonzero radical J(R) with nilpotency index m. The residue field F = R/J(R) is of order p r . We refer to [1,3,4,12,13] for the verification of the statements given here.
The ring R has a coefficient subring S = GR(p n , r) Z p n [a] for some a of multiplicative order p r − 1. If x ∈ J i (R) \ J i+1 (R) = J i , we define wt(x) = i. Let wt(π) = 1, then, J(R) = (π) and Let π ∈ J 1 be fixed, and let U(R) denote the group of units of R, then where C p r −1 =< a > is a cyclic group of order p r − 1, and H = 1 + J(R) is the p-Sylow subgroup of U(R). Moreover, π is a root of an Eisenstein polynomial g( Let u = k p−1 , where x means the greatest positive integer less than or equal x, and let H s = 1 + J s (R), s ∈ P m = {1, 2, . . . , m}. Consider the following filtering and (admissible) function: (2.6) The series (2.5) with j and the p-th power homomorphisms η s from H s /H s+1 into H j(s) /H j(s)+1 form the so called j-diagram, however, we refer to (2.5) when we mention j-diagram. We call the j-diagram in (2.5) incomplete at s if η s is not an isomorphism, and complete if all η s are isomorphisms.
Definition 2.1. We call R incomplete (complete) chain ring if the series (2.5) is incomplete at u (complete). Now, let {α i } 1≤i≤r be a representatives system in R for a basis of F over Z p . Furthermore, let f be the homomorphism f : F → F, defined by: f (α) = α p + βα. Let k = k 1 p l ((k 1 , p) = 1), λ = l + 1 and R( j) be the range of j, then H is generated by: For each s ∈ P m , let U s and U * s be subgroups of H generated by {w is } 1≤i≤r and {w is } 2≤i≤r , respectively. Hence, U s and U * s are homogeneous groups of rank r and r − 1, respectively, and of order p ν(s) , where ν(s) is the least positive integer satisfying j ν(s) (s) = m. In particular, if p > 2 or p = 2 and n ≤ 2, (2.8) Denote c =| P m \ R( j) |, then by using j-diagram, otherwise. (2.9) ). Let q and z be positive integers such that q ≥ z − 1. Then, for 0 ≤ b ≤ p q , where v p is the p-adic valuation.
All notations mentioned above have the same meanings throughout; in addition, we denote l s = min{l, ν(s)}.

Classification of finite chain rings
, R is completely determined by its invariants. Hence, in the sequel, we assume n > 1. Now, let . Let E(R) be the set of all such polynomials g(x) (Eisenstein polynomials) corresponding to the set J 1 . If σ ∈ Aut S , we denote σ(E(R)) by: σ(E(R)) = {σ(g(x)) : σ is applied to the coefficients of g(x)}. (3. 2) The group of automorphisms of S , Aut S , is cyclic of order r generated by ρ (Frobenius map) defined as: where ζ i ∈ Γ(r), the Teichmüller set of S . Furthermore, Aut (S /p i S ) Aut S (by the natural isomorphism).
If g(x) ∈ E(R), we denote R g by: Proof. Assume that R T and ψ is the isomorphism. Let ψ | S = σ and π be a root of g(x) ∈ E(R). Then, it is easy to justify that ψ(π) is a root of σ(g(x)) in T. Thus, σ(g(x)) ∈ E(T ), and so σ(E(R))∩ E(T ) φ. Conversely, if σ(g(x)) has a root θ in T for some σ ∈ Aut S and g(x) ∈ E(R). Then, the corresponding ψ( s i π i ) = σ(s i )θ i is obviously an isomorphism.

Proposition 3.2. For any finite chain rings R and T, either E(R) ∩ E(T ) = φ or E(R) = E(T ).
Proof. If E(R) ∩ E(T ) φ, then Proposition 3.1 concludes that R T and, hence, E(R) = E(T ).

Complete chain rings
Let R be a complete chain ring, then (cf. [1]) the system {w is } in (2.7) forms a basis for H, and by (Theorem 3, [3]), Let R and T be two complete chain rings with the same invariants p, n, r, k, m. Assume that π k = pβ 1 h 1 and θ k = pβ 2 h 2 for R and T, respectively. If R T, then via Proposition Observe that, σ can be considered isomorphism maps θ to ζ (proof of Proposition 3.1). Therefore, Thus, we consider the invariants, p, n − 1, r, k, m. From Eq (3.8), Since we have the same structure for H R and H T , then where h 1 = s R( j) u s and h 2 = s R( j) w s . By (2.9), there are exactly c equations of the form (3.10). Moreover, since u s , w s ∈ U s , then u s = r i=1 w a i is and w s = r i=1 (1 + α i θ s ) m i , where a i and m i are considered mod p l , i.e., a i , m i ∈ Z p l . For the converse, assume (3.7) and (3.10) hold for all s ∈ P m \ R( j) and some σ ∈ Aut S . Then, there exist β ∈< a > and δ ∈ H R such that where ψ is the corresponding: Clearly, π 1 is a root of σ(g(x)) in R and, hence, ψ is an isomorphism. Thus, the following theorem is proved.
Theorem 3.1. Let R and T be two complete chain rings with the same invariants p, n, r, k, m. Then, R T if and only if (3.7) and (3.10) hold for all s ∈ P m \ R( j). Corollary 3.3. If R is associated with p, 2, r, k, k + 1 such that (p r − 1, k 1 ) = 1. Then, R is uniquely determined up to isomorphism by its invariants.
Example 3.1. If R and T are two finite chain rings with the same invariants such that l ≥ 1 and n > 2 or n = 2 and t > 1. Assume for any σ ∈ Aut S . Hence, from Theorem 3.1, R and T are not isomorphic. By similar argument one can show R T when Theorem 3.2. Let N be the number of complete chain rings with invariants p, n, r, k, m such that n ≥ 3 or n = 2, t > 1. Then, Proof. For s R( j), U s is a homogeneous group of order p ν(s) and of rank r, and thus U s /U p l s is also a homogeneous group of order p l s and of the same rank. It follows that every u s of U s /U p l s can be written as u s = r i=1 w a i is , where the exponents taken modulo p l s . Now, to simplify notations, we can identify U s /U p l s , as a set, as Γ * (r) l s . Since there are c direct summands of H, i.e., c equations of the form (3.10), we identify H/H p l as Γ * (r) l s 1 × Γ * (r) l s 2 × · · · × Γ * (r) l sc . Moreover, replace < a > / < a k 1 > by the additive group Z z of integers modulo z, where z = (p r − 1, k 1 ). Let Aut S =< ρ > acts on the set Z z × Γ * (r) l s 1 × Γ * (r) l s 2 × · · · × Γ * (r) l sc by: (3.14) According to Theorem 3.1, it suffices to verify that N given in (3.13) is the number of equivalence classes. The number of elements fixed by ρ i is ( Therefore, Burnside Lemma computes the total number of equivalence classes.
Corollary 3.4. If l < n − 1, then l s = l, thus, where φ is Euler function and τ(a) is the order of p in Z a . Note that the right-hand side of (3.17) represents the number of finite chain rings when p k, and it was given by Clark [4].
Next, we consider a subclass which consists of all pure chain rings with the same invariants. First, we determine H p l ∩ 1 + pS . By Lemma 2.1, one can directly prove Then [1], where b s = log p u s and x is the smallest positive integer greater that or equal to x. Moreover, Lemma 3.1. Let R be a pure finite chain ring with l ≥ 1, and n > 2 or n = 2 and t > k 1 p l−1 + · · · + k 1 .
Assume that R is a pure finite chain ring with p, n, r, k, m. Then, Proof. First, if n = 2 and t ≤ k 1 p l−1 , then it is easy to see that every element of 1 + pS is given by 1 + π k δ p l , for some δ ∈ Γ * (r). This yields (1 + π k 1 δ) p l = 1 + π k δ p l , hence, Next, let n > 2 or n = 2 and t > k 1 p l−1 . If h ∈ H, there is s ∈ P m such that h ∈ H s \ H s+1 and h = 1 + π s , where ∈ U(R). We consider different cases for s.
(a) If s ≥ k and h p l ∈ 1 + pS . Then, where 1 is a unit in R. Since any element of 1 + pS is of the form 1 + π ak α a , where α a ∈ Γ * (r), thus, π j l (s) 1 = 0 or j l (s) = lk + s = qk and 1 ∈ U(S ) for some positive integer q ≥ l + 1, i.e., k | s. In either case, we obtain h p l ∈ 1 + p l+1 S = (1 + pS ) p l .
In this case, we have Note that if n = 2, then x k − pβ ∈ E(R) for every pure chain ring R, thus, N is given by (3.17). The following theorem gives N when n ≥ 3.
Theorem 3.3. The number N of pure finite chain rings with same invariants p, n, r, k, m such that n ≥ 3 is precisely Proof. The proof follows directly from Proposition 3.3 and Theorem 3.2.

Remark 3.2. Note that N in Theorem 3.3 is dependent on n.
The following corollary illustrates that the result in [7] is just a special case of Theorem 3.3.

Incomplete chain rings
In this section, we investigate the incomplete case. If R is a finite chain ring with invariants p, n, r, k and m, and π k = pβh. The incomplete situation happens when (cf. [1]), p − 1 | k, −β ∈ F * p−1 and m > k + u, (3.34) (k = (p − 1)u). However, if R is incomplete, the system (2.7) and ξ = w 1s 0 = 1 + α 1 π s 0 are subjected to where a is , a 0 are positive integers divisible by p.
If R and T are two incomplete chain rings with same invariants p, n, r, k, m. Then, H R and H T may not have the same structure [1]. This situation makes the enumeration much harder than the complete case.

Special incomplete chain rings
If we write H R (3.36) as H R = G R ⊗ s≥u 1 U s ⊗ G 2 , where G R = 1≤s≤u 1 −1 U s and u 1 = u p l . Thus, H R and H T have the same summand G R ; G T G R .
We state the following theorem without proof because it involves the same ideas to that ones of Theorem 3.2. Note that l s , in this case, equal l for 1 ≤ s ≤ u 1 − 1.
Theorem 3.4. Let Σ be the class of all finite incomplete chain rings R which have the same invariants p, n, r, k and m, and associated with π k = pβh, where h ∈ G R . If N is the cardinality (up to isomorphism) of Σ, then = h for β 1 ∈< a > and h 1 ∈ H(R 0 ), then one can check that Next, we aim to obtain the number of non-isomorphic rings in . First, we introduce some useful information about R 0 .
The proof involves argument similar to that of Proposition 3.1 with help from Corollary 3.7. Proof. The proof is conducted by induction on i. First, let i = 1, and note that H p s ⊆ H j(s) . If y ∈ H j(s) , then y = u j(s) y 1 , where u j(s) ∈ U j(s) and y 1 ∈ H j(s)+1 . Moreover, u j(s) = u p s for some u s ∈ U s , and y 1 = u j(s)+1 y 2 , where u j(s)+1 ∈ U j(s)+1 and y 1 ∈ H j(s)+2 . Since then j(s) + 2 and j(s) + 1 ∈ R( j). Which follows that u j(s)+1 = u p s 1 . Continuing in this way, we obtain y = y p 0 , and hence H j(s) ⊆ H p s . Thus, H j(s) = H p s . For i > 1, note that H p i s = (H p i−1 s ) p , and by the induction step, the result follows.
(H(R 0 )) p l , if n > 2 or n = 2 and t > k 1 p l−1 , if n = 2 and t ≤ k 1 p l−1 . (3.42) The proof follows by slightly modifying the proofs of Lemma 3.1 and Proposition 3.3, that is, consider u instead of k, and the Eq (3.35) as illustrated in (Example 2, [1]).
Theorem 3.5. Assume that l < n − 2, the number N of incomplete chain rings exist in the class is
Corollary 3.8. The number N of pure chain rings with p, n, r, k, m is precisely given in (3.33) Proof. In the case l < n − 2, the proof is just a direct application to the previous Theorem 3.5 when α = 1, ζ = 1 and N 0 = 1. If l ≥ n − 2, the proof follows from Proposition 3.4 and the proof of Theorem 3.2.

General incomplete chain rings
The general case of incomplete chain rings when n ≥ 4 or n = 3, t > 1 is still complicated to determine N. For the moment, the best we can do is to approximate N by finding upper and lower bounds. First, we derive a relation between Aut R and N.
Corollary 3.9. Assume the hypotheses of Lemma 3.5, then g 1 (x) = g 2 (x) if and only if φ ∈ Aut S R (fixing S ).
Define a relation ∼ on E(R) by: g 1 ∼ g 2 if and only if g 2 = σ(g 1 ); that is, G = Aut S acts on E(R). This relation is well defined since σ(E(R)) ⊆ E(R) (Corollary 3.1). Let orb(g) denotes the orbit of g and G g is the stabilizer of g in G. Hence, | orb(g) |= r/ | G g |. Proposition 3.6. Let R runs over all non-ismorphic classes of finite chain rings with the same invariants p, n, r, k, m. Then, (3.45) Proof. Let Aut S R acts on J 1 in the natural way, i.e., φπ = φ(π). Let ∼ be the induced equivalence relation. Thus, J 1 splits into classes of elements, and Corollary 3.10 implies that each class has | Aut S R | elements. Also Lemma 3.5 emphasizes that the number of the equivalent classes is | E(R) |. Hence, On the other hand, E(p, k) splits into non-intersecting by Proposition 3.2. Thus, Moreover, from Proposition 3.6, Therefore, N ≥ p r r .  In this case, they coincide.
The following result is easy to check so we skip the proof.  Proof. Forward from Proposition 3.7 and Corollary 3.12.

Finite chain rings and p-adic fields
In this section, we apply the above-mentioned results to the p-adic number fields. Any finite extension of Q p is called a p-adic number field where Q p is a completion of Q using p-adic norm | . | generated from p-adic valuation v p , defined as: |a| = p −v p (a) . Let K be a p-adic number field with ramification index k and residue degree r. There is a unique extension v of v p normalized such that v(π) = 1, where π is the unique prime element (uniformizer) of O K ring of integers of K.
Lemma 4.1. If L is unramified extension of degree r over Q p and K is a totally ramified extension of degree k over L. Let g(x) be an Eisenstein polynomial over O L of degree k such that its image in O L /(p n ) has a root in O K /(π m ), then if m > 2[(l + 1)k − 1], g(x) has a root in O K .
Theorem 4.1. Let N be the number of isomorphism classes of finite commutative chain rings associated with the same p, n, r, k and m, with m > 2[(l + 1)k − 1]. Then, N is the number of Q p −isomorphism finite extension of Q p with ramification index k and residue degree r.
Proof. Let K 1 and K 2 be both extensions over Q p with ramification index k and residue degree r. Then, K 1 and K 2 have the same maximal unramified extension L over Q p . Assume R = O 1 /(π m ) and T = O 2 /(θ m ), where O 1 and O 2 are the rings of integers of K 1 and K 2 , respectively. Now, if R T, then by Proposition 3.1, there is σ ∈ Aut S such that σ(g(x)) has a root in T, where g(x) is an Eisenstein polynomial over L. Assume θ is the root of σ(g(x)), and let ζ ∈ O 2 be a lifting of θ. Note that σ(g(θ)) = 0 and so τ(g(ζ)) ∈ (θ m ), where τ ∈ Aut Q p L is the corresponding to σ since Aut Q p L Aut S (L is unramified over Q p ). Since m > 2[(l + 1)k − 1], then by Lemma 4.1, f (x) = τ(g(x)) has a root π 2 in K 2 . Thus, K 1 = L(π 1 ) L(π 2 ) = K 2 , (4.3) where π 1 is a root of g(x) in K 1 . Also note that π 2 ζ mod θ. This ends the proof.
The following example shows that the condition on m in Theorem 4.1 is necessary.

Conclusions
In this paper, we have investigated the classification of finite commutative chain rings with the same invaraints p, n, r, k, m. If (p − 1) k, the full classification of these rings is given. While if (p − 1) | k, we showed that the number of non-isomorphic classes of finite commutative chain rings depends not only on their invariants but also on their Eisenstein polynomials. In this case, we classified such rings up to isomorphism under some conditions concerning the Eisenstein polynomials.