Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring

: A ﬁnite ring with an identity whose lattice of ideals forms a unique chain is called a ﬁnite chain ring. Let R be a commutative chain ring with invariants p , n , r , k , m . It is known that R is an Eisenstein extension of degree k of a Galois ring S = GR ( p n , r ) . If p − 1 does not divide k , the structure of the unit group U ( R ) is known. The case ( p − 1 ) | k was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of U ( R ) . In this article, we manage to determine the structure of U ( R ) when ( p − 1 ) | k by ﬁxing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order.


Introduction
In this article, we consider finite commuative chain rings, although some results still correct under more general situation. Ayoub called these rings (cf. [1]) primary homogeneous rings. Such rings arise in various places; in Coding Theory (cf. for example [2,3]); in Geometry (cf. [4]). It is well-known that every finite chain ring R is an Eisenstein extension of some degree k over a Galois ring of the form S = GR(p n , r). There are positive integers p, n, r, k and m associated with R, called invariants of R. Our main aim in this study is to obtain the structure of the multiplicative group U(R) of R. In (1972) (cf. [1]), Ayoub obtained various results based on ideas (cf. [5]) regarding j-diagrams for abelian p-groups.
One major result claims that the factorization of an abelian p−group with incomplete j-diagram can be completely obtained by the mentioned diagram. This idea was then used to find the structure of U(R) when R is not necessarily finite (Theorem 3, Section 4 [1]). However, later, it turned out that incomplete j-diagrams failed to determine the exact decomposition of multiplicative groups of some examples introduced by Luis [6]. On the other hand, Hou [7] by different approach gave the structure of U(R) in a special case; p k. This sudy demonstrates that the result (Theorem 3, [1]) is still valid in some cases, for instance the case studied in [7]. Additionally, we enhance the idea that incomplete j-diagrams are generally not enough to obtain the decomposition of bounded p-groups. That is, there is a relation amongst the generators which plays essential role in determining the structure. That relation depends not only on the related incomplete j-diagram, but also on the Eisenstein polynomial by which R is constructed over its coefficient subring GR(p n , r). However, with this relation being taken into consideration, we manage to make incomplete j-diagrams succeed in recapturing the structure of multiplcative groups of finite chain rings. In addition, a set of linearly independent generators for U(R) is provided and, further, the number of such generators is computed for each possible order. Finally, we give the correct version of ( [5], p. 458).

Preliminaries and Notations
In what follows, R will denote a finite commutative chain ring of characteristic p n and nonzero radical N (the case when N = 0, R is a field) with index of nilpotency m. The residue field F = R/N is a finite field of order p r . We now state some facts and introduce notations that we shall use throughout. For the details, we refer the reader to [8] for finite chain rings and [1,5] for j-diagrams.
The ring R contains a subring (coefficient subring) S of the form S = GR(p n , r) ∼ = Z p n [a], where a is an element of S of multiplicative order p r − 1. If π ∈ N \ N 2 , it is easy to see that N = (π), since N/N 2 is of 1 dimension over F. In this case, R is expressed as: (as S−module) where k is the greatest integer i ≤ m such that p ∈ N i . It follows that where s i ∈ S. This means that π is a root of an Eisenstein polynomial over S of the form: The positive integers p, n, r, k and m are called invariants of R. Furthermore, m = (n − 1)k + t, for some 1 ≤ t ≤ k. Let H s = 1 + N s , s ∈ P m = {1, 2, . . . , m} and consider the following filtering: joined with a function j defined as: where k = (p − 1)u + v, 0 ≤ v < p − 1. Note that if Char(R) = p, i.e., n = 1, put k = m. The series (4) with j defined in (5) and the p-th power homomorphisms η s from H s /H s+1 into H j(s) /H j(s)+1 form what we call j-diagram. However, we refer to the series (4) when we mention j-diagram. If there exists m = s ∈ P m , such that η s is not an isomorphism, we then call the series (4) incomplete j-diagram at s and complete j-diagram otherwise (all η s are isomorphisms). We denote, by U(R), the units group of R, and so one can verify that where < a > is a cyclic group of order p r − 1 and H = 1 + πR is the p-Sylow subgroup of U(R). The structure problem of U(R) is then reduced to that of H. After Ayoub [1] we call H the One Group of U(R). If N m (e) is the number of basis elements in H whose orders p e , then H is decomposed as: where C p e is a cyclic group of order p e . We define p-rank N m of H as If π ∈ N\N 2 fixed, then from Eisenstein polynomial (3) where β ∈< a > and h ∈ H. Now, let {α i } 1≤i≤r be a representatives system in R for a basis of F over its prime field ( Z p ). Set where R(j) is the range of j. For each s ∈ P m , let U s be a subgroup of H generated by {w is } 1≤i≤r . Hence one can easily show that H s = U s × H s+1 . Now, if (4) is a complete j−diagram, then, from (cf. [1]), the system {w is } ( * ) forms a basis for H. (4) is a complete j-diagram. Then, The purpose of the present paper is to establish a basis for H when (4) is an incomplete j-diagram at u. Moreover, N m (e) and N m are determined in both cases (complete and incomplete) in Theorems 3 and 4.
Unless otherwise mentioned, all of the symbols stated above will retain their meanings throughout and, additionally, if s ∈ P m , ν m (s) is the least positive number fulfilling j ν m (s) (s) = m.

The Determination of the Structure of H
Throughout this section, k = k 1 p l , where (k 1 , p) = 1 and l ≥ 0. The following proposition is useful and it will be needed later.

Proposition 2.
If m > k + u, then the following conditions are equivalent: Proof. Assume that (i) holds, then ker η u = 1 since η u is onto. Then there is 1 + απ u ∈ ker η u , such that (1 + απ u ) p = 1 + α p π up − βαπ u+k ξ = 1 mod H up+1 , where ξ ∈ H. This only happens when pu = u + k and α p − βα = 0 mod p. Hence, if (p − 1) | k, ker η u is isomorhpic to that of the homomorphism, f : However, ker f = 1 if and only if x p − βx has nonzero solutions in F. The remaining hypotheses follow immediately. Definition 1. We call R incomplete (complete) chain ring if one of the equivalent conditions in Proposition 2 holds (otherwise).

Corollary 1.
If R is an incomplete chain ring, then ker η u is of rank p.

Remark 1.
In the light of Proposition 2 and its corollary, v = 0 and k 1 = (p − 1)s 0 , i.e., u = s 0 p l . Furthermore, if η is the composition of η s 0 , . . . , η u then ker η is of rank p. Let be a generator of ker η.
Assume that f is as mentioned in the proof of Proposition 2 and λ = l + 1. Construct the following system: The following proposition is stated without proof, since it involves the same ideas of that in the complete case.

Proposition 3.
The set {w is , γ} ( * * ) represents a system of generators for H.
Observe that ξ = w 1s 0 ∈ ker η and then where a is and a 0 are positive integers that are divisible by p, i.e., µ ≥ 1. Note that µ will keep its meaning throughout the paper, as described here. Let H 0 be a subgroup of H that is generated by {γ, w is } (i,s)∈B . Thus, by (cf. [1]), H 0 has a direct decomposition, Theorem 1. Let R be an incomplete chain ring and λ ≤ µ, then where D is a cyclic group of order p λ .
Proof. By Equation (13), it is clear that H 0 ∩ < ξ >=< ξ p λ > and also H 0 is a p-pure subgroup of H; every element is not of a smaller p-height in H 0 than in H. Thus, by the results from (cf. [9]), H 0 is a direct summand of H. This finishes the proof.

Remark 2.
(1) It is worthy to mention that Theorem 1 coincides with Ayoub's results (Theorem 3 Section 4, [1]). That means in such case, incomplete j-diagrams succeed in retrieving the structure of H. (2) If p k, then obviously λ = 1 ≤ µ and, thus, the result that is given in [7] is just a particular case of Theorem 1.
From now on, we assume λ > µ. Consider Equation (13) and let Lemma 1. If R is an incomplete chain ring and λ > µ, then there is a positive integer d and p-pure Proof. For (i, s) ∈ A, let a is = v p (a is ), where v p is the p-adic valuation. Subsequently, it is clear that µ ≤ a is for all (i, s) ∈ A. Choose (i 1 , s 1 ) ∈ A such that a i 1 s 1 = µ. If there are more than one of s satisfying a is = µ, choose the smallest one. Let h is be defined as a is = h is p a is , (h is , p) = 1. If B 1 = A \ {(i 1 , s 1 )}, h 1 = h i 1 s 1 and w 1 = w h 1 i 1 s 1 , then Equation (13) can be rewritten as: w where τ = ∏ (i,s)∈B 1 w δ is is , δ is = −h is p a is −µ 1 and f 1 = min{a is : (i, s) ∈ B 1 }. Define G 1 to be the subgroup of H that is generated by {w is , ξ} (i,s)∈B 1 . Thus, from (17), Because µ ≤ f 1 , then the p-height of w 1 in G 1 is not lesser than that in H and thus G 1 is a p-pure subgroup of H. Now, if we raise Equation (17) to p ν(s 1 )−µ , we obtain a new relation among the generators of G 1 , where b is = c is p a is +ν(s 1 )−µ , and B 1 ⊆ B 1 , because some generators might disappear due to raising (17) to a power of p. Note that since the generators on the right side of Equation (18) are linearly independent, thus the left side cannot vanish. Again, there is (i 2 , s 2 ) such that a i 2 s 2 = a 2 ≥ µ is the smallest a is ((i, s) ∈ B 1 ) and then , then we have a similar situation to that of w 1 . Let B 2 = B 1 \ {(i 2 , s 2 )} and G 2 be a subgroup of G 1 generated by ξ and all w is with (i, s) ∈ B 2 . By a similar arguement, G 2 is a p-pure subgroup of G 1 and, thus, transitively (Lemma 26.1, [9]) G 2 is a p-pure subgroup of H. Therefore, after a finite number d of similar processes, we obtain: Remark 3. Based on notations that were introduced in Lemma 1, put Ω = {s 1 , . . . , s d } and let where Λ = {P m \ R(j)} \ Ω and Ω = Ω ∪ {s 0 }.

Theorem 2.
Let R be an incomplete finite chain ring with invariants p, n, r, k, m such that λ > µ. Then, (1) U s is a direct product of r cyclic groups of order p ν(s) .
(ii) U * s i is a direct product of r − 1 cyclic groups of order p ν(s i ) , and D i is a cyclic group of order p µ i . (iii) D is a cyclic group of order Proof. Because, for each 1 ≤ i ≤ d − 1, G i+1 is a p-pure subgroup of G i (Lemma 1) and also a bounded p-group, then (Theorem 27.5, [9]) G i+1 is a direct summand of G i ; there is a subgroup L ⊆ G i , such that G i = G i+1 ⊗ L. Thus, based on (20) and Remark 3, By the argument of the proof of Lemma 1, where D i is a cyclic group of order p µ i . Therefore, the decomposition (22) is just a result of successive application to the above argument. Note that D =< ξ > .

Corollary 4.
With the same assumption in Theorem 2, N m = cr + 1.

Corollary 6.
Let R and T be two incomplete chain rings with same invariants p, n, r, k, m and same Ω and {µ i } 1≤i≤d . Then, H R ∼ = H T .
In general, Lemma 1 gives an algorithm for calculating Ω and µ i s. The following example illustrates that algorithm. Example 2. Let R be an incomplete chain ring with invariants p, n, r, k, m and π k = pβh, where h ∈ H(R 0 ) and R 0 ⊂ R is an Eisenstein extension over S by x p−1 + p, i.e., R 0 = S[π 0 ], π 0 is a root of x p−1 + p (Proposition 2). Clearly, R 0 is an incomplete chain ring with p, n, r, p − 1, m 1 , where m 1 = (n − 1)(p − 1) + t 1 and t 1 = t k 1 p l . Consider the (admissible) function j 0 with A similar equation of that in (13), where ξ 0 = 1 + π 0 . Let R 1 = R 0 [π 1 ] be an Eisenstein extension over R 0 by x p − π 0 h 1 , where h p−1 1 = h. R 1 is an incomplete chain ring with invariants p, n, r, p, m 2 , such that m 2 = pm 1 . Now, put π 0 = π p 1 h −1 1 in (26) and after forward computations we get where ξ 1 = 1 + π 1 , y ∈ H s (R 1 ) p and s > s 1 = p(p − 1) + 1. If j(s 1 ) < m 1 , then based on Lemma 1 Ω 1 = {s 1 }. Continuing in this way and after l steps (k = s 0 (p − 1)p l ), R is an Eisenstein (b) In the case when u > t. Firstly, observe that ν m 0 (s) = 1 = e if and only if s ≥ k + t. By the same reasoning of Case (a), we obtain the results. Secondly, when e > ι, in this case all s satisfying ν m 0 (s) > 1 occur in the interval s < t if u ≤ t or s < k + t when t < u. Therefore, we return to a similar situation of Theorem 3 with m 0 − ak and e − a instead of m 0 and e, respectively. Observe that, if ι = 1 and e = λ + 1, then there exist r − 1 generators of U s 0 and then put z = −1. When e = λ, we have to add z = 1, since o(ξ) = λ. On the other hand, if ι = µ, there are z e of U s each having r − 1 elements of basis of the same order, so put z = −z e .
In conclusion, this article considered the multiplicative groups of finite commutative chain rings. The case when (p − 1) | k was completed for every incomplete chain ring. Moreover, a set of linearly independent generators for H was established by connecting those generators to a basis of F over its prime field.
Recently, there has been increasing interest in using finite commutative chain rings in Coding Theory. Researchers may wish to further apply this knowledge about finite commutative chain rings to Coding Theory.