In this paper, we present some basic algorithms for manipulation of
finitely generated modules over finite chain rings. We start with an
algorithm that generates the standard form of a matrix over a finite chain
ring, which is an analogue of the row reduced echelon form for a matrix over
a field. Furthermore we give an algorithm for the generation of the union of
two modules, an algorithm for the generation of the orthogonal module to a
given module, as well as an algorithm for the generation of the intersection
of two modules. Finally, we demonstrate how to generate all submodules of
fixed shape of a given module.

ACM Computing Classification System (1998): G.1.3, G.4

Georgieva, Nevyana

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 10 (2016), No 3–4, 285–297 SerdicaJournal of ComputingBulgarian Academy of SciencesInstitute of Mathematics and InformaticsBASIC ALGORITHMS FOR MANIPULATIONOF MODULES OVER FINITE CHAIN RINGS∗Nevyana GeorgievaAbstract. In this paper, we present some basic algorithms for manipula-tion of finitely generated modules over finite chain rings. We start with analgorithm that generates the standard form of a matrix over a finite chainring, which is an analogue of the row reduced echelon form for a matrix overa field. Furthermore we give an algorithm for the generation of the union oftwo modules, an algorithm for the generation of the orthogonal module to agiven module, as well as an algorithm for the generation of the intersectionof two modules. Finally, we demonstrate how to generate all submodules offixed shape of a given module.1. Introduction. The importance of the class of linear codes over finitechain rings was recognized only in the last two decades. The interest in thesecodes was initiated by the remarkable discovery that some notorious classes ofACM Computing Classification System (1998): G.1.3, G.4.Key words: chain rings, finitely generated modules over finite chain rings, the orthogonalmodule, linear codes over finite chain rings, standard form of a matrix over a chain ring.*This research was partially supported by the Scientific Research Fund of Sofia University.286 Nevyana Georgievacodes like the Kerdock- and Preparata-codes outperform the classical linear codeswith the same parameters [1, 2]. Attempts were made to develop a general theoryof linear codes over finite chain rings. This was done in a series of papers byHonold and Landjev [3, 4, 5, 6]. It turned out that linear codes over finite chainrings are equivalent to multisets of points in special geometries, the so-calledprojective Hjelmslev geometries [7, 8, 9]. The efforts of various researches werefocused on proving analogs for all important results known from finite geometry.However, the situation in the projective Hjelmslev geometries turned out to befar more unclear due to their complicated internal structure. This has to do withthe more complex structure of finitely generated modules over finite chain ringscompared with vector spaces over finite fields. In some cases even the constructionof examples turns out to be a problem. For instance, it is difficult to constructR-spreads of mutually non-neighbouring lines in PHG(R4) (cf. [10]). This makescomputer searches unavoidable and explains the need for efficient algorithms thatdeal with finitely generated modules over finite chain rings.The paper is structured as follows. In Section 2 we give some basic factsconcerning the structure of finite chain rings and the representation of the ele-ments of a finite chain ring. Further we give a structure theorem for modules andresults about the orthogonal module, and explain notions like the shape and thedual shape of a module. In Section 3 we define the standard form of a matrixand describe an algorithm that puts a matrix into standard form. Furthermore,this algorithm is applied to get algorithms for the generation of the union of twomodules and for containment of a given module in another module. Section 4 isdevoted to the generation of the orthogonal module M⊥R to a given module RM .We use an explicit form for a matrix whose rows generate M⊥R that was recentlyfound in [11]. This algorithm in turn allows us to find the intersection of twomodules. Finally in Section 5 we describe an algorithm that generates all sub-modules of given shape for a module defined by the rows of a matrix in standardform.2. Basic facts. In this section we describe briefly some basic propertiesof finite chain rings and finitely generated modules over finite chain rings. For amore in-depth introduction to this subject we refer to [12, 13, 14].An associative ring with identity is called a left (right) chain ring if thelattice of its left (right) ideals is a chain. In such case, there exists an elementθ ∈ RadR \ Rad2R whose powers generate all idealsR > (θ) > (θ2) > · · · > (θm−1) > (θm) = (0).Basic algorithms for manipulation of modules . . . 287The smallest m such that θm = 0 is called the length (or nilpotency index) of R.The field R/(θ) is called the residue field of the ring R. If we set |R/(θ)| = q, wehave |R| = qm. Let Γ = {γ0 = 0, γ1 = 1, γ2, . . . , γq−1} be a set of elements of Rwith γi 6≡ γj (mod RadR) for all i, j with 0 ≤ i ≤ j ≤ q−1. For every element afrom R there exists a unique representationa = a0 + a1θ + · · ·+ am−1θm−1, ai ∈ Γ.We fix some linear order on Γ:γ0 ≺ γ1 ≺ . . . ≺ γm−1.This order is extended to all elements of R as follows. For two elements a =a0 + a1θ+ . . .+ am−1θm−1 and b = b0 + b1θ+ . . .+ bm−1θm−1 from R, ai, bi ∈ Γ,we write a ≺ b if and only ifam−1 = bm−1, . . . , aj+1 = bj+1, aj ≺ bj ,for some 0 ≤ j ≤ m − 1. This is in fact the lexicographic order on the n-tuples(a0, . . . , am−1). We can define a bijection ϕ : R → {0, 1, . . . , qm − 1} which isconsistent with the linear order of the elements of R given above. Set ϕ(γi) = i.Further for a = a0 + a1θ + . . .+ am−1θm−1, ai ∈ Γ, we let ϕ(a) =m−1∑i=0ϕ(ai)qi.In [11] we proved the following lemma.Lemma 1.(1) The element a is in Γ if and only if ϕ(a) < q; more generally, the ele-ments a with ϕ(a) < qi form a system of distinct representatives modulo(RadR)i;(2) for each i ∈ N, a ∈ (RadR)i, i. e., a = bθi, b ∈ R, if and only if ϕ(a)divides qi;(3) if qi divides ϕ(b) then b = aθi with a = ϕ−1(ϕ(b)qi).The next theorem is the main structure result for finitely generated mod-ules over finite chain rings.288 Nevyana GeorgievaTheorem 1. Let R be a finite chain ring. For every finite module RMthere exists a uniquely determined partition λ = (λ1, . . . , λk) ` logR |M | into parts1 ≤ λi ≤ m such thatRM ∼= R/(RadR)λ1 ⊕ . . .⊕R/(RadR)λk .The parts of the conjugate partition λ′ = (λ′1, λ′2, . . .) ` logq |M | are the Ulm–Kaplansky invariants λ′i = dimR/RadR(M [θ] ∩ θi−1M).The partitions λ and λ′ are called the shape and conjugate shape of RM .The integer k = λ′1 = dimR/RadRM [θ] is called the rank of RM and the integer λ′mis called the free rank of RM .Let RM be a finitely generated left R-module. We say that the elementx ∈ RM has period θi if i ≥ 0 is the smallest integer with θix = 0. The set ofall elements of period m are called torsion elements. We also set M∗ = {x ∈M | x has period θm}. The element x has height j if j is the largest integer withx = θjy, y ∈M∗.3. The standard form. In this section we introduce the notion ofstandard form of a matrix over a finite chain ring. It is analogous to the notionof row-reduced echelon form of a matrix over a field. Denote by Mk,n(R) the setof all matrices over the finite chain ring R. The following definition was givenin [11].Definition 1. The matrix A = (aij) ∈ Mk,n is said to be in standardform if(1) aiji = θm−ti for some ti ∈ {0, . . . ,m};(2) ais = θm−ti+1β, β ∈ R, for all s < ji;(3) ais = θm−tiβ, β ∈ R, for all s > ji;(4) asji ≺ aiji for all s 6= i (here ≺ is the lexicographic order defined insection 2);(5) i1 < i2 < i3 < . . ..The integer ti is called the type of row i, i = 1, . . . , k. Let a = (a1, . . . , an) ∈RRn. The smallest i ∈ {0, . . . ,m} such that θia = 0 is called the type of a. Theleftmost component aj with aj ∈ (RadR)m−i \ (RadR)m−i+1 is called the leaderBasic algorithms for manipulation of modules . . . 289of a. For a matrix A ∈Mk,n(R) in standard form we denote the set of coordinatepositions of the row-leaders of A, i. e., J(A) = {j1, j2, . . . , jk}. Note that in thedefinition we decided to suppress the zero-rows. In [11] the following two resultswere proved.Theorem 2. Let RM ≤ RRn be a module and let A be a matrix in stan-dard form whose rows generate RM . For an arbitrary element v ∈ RM denoteby s the position of its leader. Then s ∈ J(A).Theorem 3. For every module M ≤ RRn there exists a unique matrix Bin standard form such that M is spanned by the rows of B.Corollary 1. Let A be a (k × n)-matrix in standard form over the chainring R. There exist permutation matrices T1 of size (k×k) and T2 of size (n×n)such that(1) T1AT2 =Ik0 A01 A02 . . . A0,m−1 A0,m0 θIk1 θA12 . . . θA1,m−1 θA1,m0 0 θ2Ik2 . . . θ2A2,m−1 θ2A2,m.......... . .......0 0 0 . . . θm−1Ikm−1 θm−1Am−1,m ,where the entries in the matrices Aij are from R.Let a matrix M generating the module RM be given. The followingalgorithm generates a matrix in standard form, whose rows generate RM .Algorithm 1: The standard form of a matrixInput: a k × n matrix M with entries from ROutput: a matrix A in standard form whose rows generate the samemodule as the rows of M1: for t = m, . . . , 1 do2: for every row r of the matrix M do3: if not all components of r are multiples of θm−t+1 then4: Find i the leftmost position i of r that is not a multiple of θm−t+1290 Nevyana Georgieva5: Left multiply all components of r by(ϕ−1(ϕ(ri)qm−t))−16: Set the matrix to be B = A ∪M \ {r}//B is the union of the rows of A and M with r excluded7: for every row r′ of B do8: Let c = ϕ−1(bϕ(ri)qm−tc)and replace r′ by r′ − c · r9: if r′ = 0, remove it from M10: endfor11: Put r as the ith row of A and remove it from M12: endif13: endfor14: endfor15: for every row a of the matrix A16: Let j be the leftmost position in a that is > 017: Let b = A \ {a} be a row of A preceding a18: if bj ≥ aj do19: Set c1 = dqm − bjaje and replace b by b+ c1 · a20: endif21: endfor22: return AAlgorithm 1 can be used in an obvious way to generate the union of twosubmodules. We just consider a new matrix consisting of the generator matricesof the two given modules and put it in standard form by Algorithm 1.Basic algorithms for manipulation of modules . . . 291Algorithm 2: Union of submodulesInput: two matrices M and N of sizes k1-by-n and k2-by-nOutput: a matrix A in standard form whose rows generate the modulespanned by the rows of M and N1: Let P =(MN)2: Apply Algorithm 1 to get the standrard form U of the matrix P3: return UNow we can also easily test whether a submodule is contained in a givenmodule. This is done by the following algorithm.Algorithm 3: Submodule-testInput: matrices A and B in standard form with the same number ofcolumnsOutput: Yes if the submodule spanned by the rows of B is contained inA; No otherwise1: Create the matrix C =(AB)2: Generate the matrix C1 which is the standard form of C3: If C1 = A then B is contained in A;4: print: Yes5: else print: No4. The orthogonal submodules. Let R be a finite chain ring andconsider a left module RM ≤ RRn. For two vectors x = (x1, . . . , xn) and y =(y1, . . . , yn) we define their inner product byxy = x1y1 + . . .+ xnyn.292 Nevyana GeorgievaThe right orthogonal to RM is defined byM⊥R = {y ∈ Rn | xy = 0 for all x ∈M}.Note that the orthogonal to a left module is a right module and vice versa. Thefollowing well-known theorem summarizes some basic properties of the orthogonalmodule (cf. [5, 6]).Theorem 4. Let R be a chain ring with |R| = qm, R/RadR ∼= Fq, andlet RM ≤ RRN be a left submodule of shape λ = (λ1, . . . , λn).(1) The right module M⊥R has shape λ = (m−λn, . . . ,m−λ1). In particular|M ||M⊥| = |Rn|.(2) ⊥(M⊥) = M .(3) M → M⊥ defines an antiisomorphism between the lattices of left (resp.right) submodules of Rn and hence(M1 ∩M2)⊥ = M⊥1 +M⊥2 , (M1 +M2)⊥ = M⊥1 ∩M⊥2 ,for M1,M2 ≤ Rn.Theorem 5. Let RM be a submodule of RRn generated by the rows of thematrix A of the form (1). Then M⊥R is generated by the matrix(2) B =B01θm−1 Ik1θm−1 0 0 . . . 0B02θm−2 B12θm−2 Ik2θm−2 0 . . . 0B03θm−3 B13θm−3 B23θm−3 Ik3θm−3 . . . 0............. . ....B0,m−1 B1,m−1 B2,m−1 B3,m−1 . . . Ikm−1 ,where(3) Bij = −(Aij −∑1<k<j+1AikAk,j+1 +∑i<k<l<j+1AikAklAl,j+1 − . . .+(−1)j−i+1Ai,i+1Ai+1,i+2 . . . Aj,j+1)T .Corollary 2. Let A ∈Mk,n be a matrix over a chain ring R whose rowsgenerate the module RM . Let A′ = T1AT2, where T1 and T2 are permutationmatrices of orders k and n, respectively, be of the form (1). The module M⊥R isgenerated by the rows ofB = T T1 B′T T2 ,where B′ is the matrix given by (2).Basic algorithms for manipulation of modules . . . 293Theorem 5 gives an explicit expression for a set of generators of the or-thogonal module.Algorithm 4: The orthogonal module M⊥Input: a matrix A whose rows generate a left module RMOutput: a matrix C whose rows generate the right module M⊥R that isorthogonal to A1: Find permutation matrices T1 and T2 that transform A to the form (1)2: Compute the matrices Bij by (3)3: Compute the matrix B by (2)4: C = T T1 BTT25: return CThe intersection of two modules is obtained from Theorem 4(3) which gives thatM1 ∪M2 = ⊥(M⊥1 +M⊥2 ). Hence we have the following algorithm.Algorithm 5: The intersection of two modulesInput: matrices A and B in standard form whose rows generate themodules M and N , respectivelyOutput: a matrix C generating the intersection of A and B1: Use Algorithm 4 to find matrices A⊥ and B⊥ which generate the right ortho-gonal modules to A and B, respectively.2: Set U =(A⊥B⊥)3: Apply Algorithm 4 to find a matrix ⊥U—left orthogonal to the matrix U4: return C = ⊥U294 Nevyana Georgieva5. Generation of all submodules of fixed shape. Let RM 5R Rnbe a module of shape λ, whereλ = (m, . . . ,m︸ ︷︷ ︸k0,m− 1, . . . ,m− 1︸ ︷︷ ︸k1, . . . , 1, . . . , 1︸ ︷︷ ︸km−1) = (λ1, λ2, . . . , λk), k =m−1∑i=0ki.In this section we present an algorithm that generates all submodules RNof RM of shape µ = (µ1, µ2, . . . , µl) where µ ≺ λ. Here l 5 k and we suppressthe trailing zeros. We assume that the module RM is generated by the rows of ak ×m matrix A which is in the form (1). Let RN be generated by the rows of al × n matrix B in standard form. The rows of B are linear combinations of therows of A. Then there exists a matrix C (also in standard form) such thatB = CA.Moreover C has the following properties:(1) if the leader in row i of B is in position ji then the leader of row i in Cis in position li = ji;(2) the components of C contained in the jth column wherek0 + · · ·+ ks−1 + 1 ≤ j ≤ k0 + · · ·+ ks−1 + ks, k−1 = 0are from Γ + θΓ + · · ·+ θm−s−1Γ.Thus the generation of all submodules of RM is equivalent to the generation of allmatrices C with properties (1) and (2). Unfortunately, the shape of the modulegenerated by the rows of B does not follow immediately from the shape of thematrix C, so we have to check this in a separate step.Algorithm 6: The generation of all submodulesInput: a matrix A in standard form whose rows generate a module ofshape λ = (λ1, . . . , λk);a fixed shape µ = (µ1, . . . ,muk) ≺ λOutput: a set of matrices whose rows generate all submodules of M ofshape µ.Basic algorithms for manipulation of modules . . . 2951: Find permutation matrices T1 and T2 to transform A to A′ which is of theform (1)2: for every k-tuple {j1, . . . , jl} ⊆ {1, . . . , k} do3: for every l-tuple (t1, . . . , tl), ti ∈ {0, . . . ,m− 1}4: Set ciji = θti , i = 1, . . . , l5: for every choice of the remaining elements of C subject to (1) and (2)6: construct the matrix C7: if shape of CA is µ then print: C8: construct all matricesLet us remark that the check of the shape in step [7 :] can be avoided ifwe generate in step [3 :] only such l-tuples that yield a submodule of shape µ.Example 1. Let R = Z4 and let A be the matrixA =1 0 1 1 20 1 1 0 10 0 2 0 00 0 0 2 2 .The module M generated by the rows (but also by the columns) of A is of shapeλ = (2, 2, 1, 1). We want to generate all submodules N ≤ M of shape µ = (2, 1).Their number is 84. We are going to construct the possible matrices C that yieldall subodules of shape µ. The matrix A is already in the required form so we skipstep [1 :]. In step [2 :] we generate all possible positions for the leaders.( • ◦ ◦ ◦◦ • ◦ ◦),( • ◦ ◦ ◦◦ ◦ • ◦),( • ◦ ◦ ◦◦ ◦ ◦ •)( ◦ • ◦ ◦◦ ◦ • ◦),( ◦ • ◦ ◦◦ ◦ ◦ •),( ◦ ◦ • ◦◦ ◦ ◦ •)In step [3 :] we try all possibilities for the leaders. Some of them will give rise tosubmodules that are not of shape µ. Using the remark before this example we have296 Nevyana Georgievathe following possibilities for the leaders themselves.(1 ◦ ◦ ◦◦ 2 ◦ ◦),(2 ◦ ◦ ◦◦ 1 ◦ ◦),(1 ◦ ◦ ◦◦ ◦ 1 ◦)(1 ◦ ◦ ◦◦ ◦ ◦ 1),( ◦ 1 ◦ ◦◦ ◦ 1 ◦),( ◦ 1 ◦ ◦◦ ◦ ◦ 1)Now we have the following possibilities to complete the matrix C. Note that thelast two columns can contain only entries from Γ = {0, 1}.(1 Γ Γ Γ0 2 0 0),(2 0 0 00 1 Γ Γ),(1 R 0 Γ0 RadR 1 Γ)(1 R Γ 00 RadR 0 1),(RadR 1 0 ΓRadR 0 1 Γ),(RadR 1 Γ 0RadR 0 0 1)Here R = {0, 1, 2, 3}, Γ = {0, 1}, and RadR = {0, 2}. Thus the number ofmatrices C of the first type is 8; of the second type, 4; of the third type, 32; of thefourth type, 16; of the fifth type, 16; and of the sixth type, 8. This gives a total of8 + 4 + 32 + 16 + 16 + 8 = 84,as already obtained.REFERENCES[1] Nechaev A. A. Kerdock code in a cyclic form. Discrete Mathematics andApplications, 1 (1991), No 4, 365–384.[2] Hammons A. R., Jr., P. V. Kumar, A. R. Calderbank,N. J. A. Sloane, P. Sole´. The Z4-linearity of Kerdock, Preparata,Goethals and related codes. IEEE Trans. on Inf. Theory, 40 (1994), No 2,301–319.[3] Honold Th., I. Landjev. Linear codes over finite chain rings. In: Proc. ofthe International Workshop on Optimal Codes, Sozopol, 1998, 116–126.[4] Honold Th., I. Landjev. Linearly representable codes over finite chainrings. Abh. Math. Semin. Univ. Hamburg, 69 (1999), 187–203.Basic algorithms for manipulation of modules . . . 297[5] Honold Th., I. Landjev. Linear codes over finite chain rings. ElectronicJournal of Combinatorics, 7 (2000), #R11.[6] Honold Th., I. Landjev. Linear codes over finite chain rings and finiteprojective geometries. In: P. Sole´ (ed.). Codes over Rings, World Scientific,2009, 60–123.[7] Honold Th., I. Landjev. Projective Hjelmslev geometries. In: Proc. ofthe International Workshop on Optimal Codes, Sozopol, 1998, 97–115.[8] Honold Th., I. Landjev. Arcs in projective Hjelmslev planes. DiscreteMathematics, 11 (2001), No 1, 53–70. Originally published in: DiskretnayaMatematika, 13 (2001), 90–109 (in Russian).[9] Landjev I. On blocking sets in projective Hjelmslev planes. Advances inMathematics of Communications, 1 (2007), No 1, 65–81.[10] Kiermaier M., I. Landjev. Designs in projective Hjelmslev spaces. In:M. Lavrauw et al. (eds). Theory and Applications of Finite Fields. Contem-porary Mathematics, 579 (2012), 111–122.[11] Georgieva N., I. Landjev. On the representation of modules over finitechain rings. Annuaire de l’Universite´ de Sofia “St. Kliment Ohridski”, 2017(submitted).[12] Nechaev A. A. Finite principal ideal rings. Math. USSR-Sb., 20 (1973),No 3, 364–382.[13] McDonald B. R. Finite Rings with Identity. Marcel Dekker, New York,1974.[14] Lam T.-Y. Lectures on Modules and Rings. Graduate Text in Mathematics,189. Springer Science & Business Media, 1999.Nevyana GeorgievaNew Bulgarian University21, Montevideo St.1618 Sofia, Bulgariae-mail: nevyanag@fmi.uni-sofia.bgReceived March 15, 2017Final Accepted May 16, 2017

Basic Algorithms for Manipulation of Modules over Finite Chain Rings

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Basic Algorithms for Manipulation of Modules over Finite Chain Rings

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