aa r X i v : . [ m a t h . R A ] M a y A Characterization of Modules with Cyclic Socle
Ali Assem
Department of [email protected]
September 24, 2018
Abstract
In 2009, J. Wood [15] proved that Frobenius bimodules have the extension property for sym-metrized weight compositions. Later, in [9], it was proved that having a cyclic socle is sufficientfor satisfying the property, while the necessity remained an open question.Here, landing in Midway, the necessity is proved, a module alphabet R A has the extension propertyfor symmetrized weight compositions built on Aut R ( A ) is necessarily having a cyclic socle. Note:
All rings are finite with unity, and all modules are finite too. This may be re-emphasized insome statements. The convention for functions is that inputs are to the left.
A (left) linear code of length n over a module alphabet R A is a (left) submodule C ⊂ A n . A hasthe extension property (EP) for the weight w if for any n and any two codes C , C ⊂ A n , anyisomorphism f : C → C preserving w extends to a monomial transformation of A n . In 1962,MacWilliams [6] proved the Hamming weight EP for linear codes over finite fields; in 1996, H. Wardand J. Wood [11] re-proved this using the linear independence of group characters. This kind of proofs– using characters – led to further generalities. In 1997, J. Wood [12] proved that Frobenius rings havethe EP for symmetrized weight compositions (swc), and in his 1999-paper [13], proved that Frobeniusrings have the property for Hamming weight. Besides, for the last case, a partial converse was proved:commutative rings satisfying the EP for Hamming weight are necessarily Frobenius.In 2004, Greferath et al.[7] showed that Frobenius bimodules do have the EP for Hamming weight.In [2], Dinh and López-Permouth suggested a strategy for proving the full converse. The strategy hasthree parts. (1) If a finite ring is not Frobenius, its socle contains a matrix module of a particulartype. (2) Provide a counter-example to the EP in the context of linear codes over this special module.(3) Show that this counter example over the matrix module pulls back to give a counter exampleover the original ring. Finally, in 2008, J. Wood [14] provided the main technical result for carryingout the strategy, and thereby proving that rings having the EP for Hamming weight are necessarilyFrobenius. The proof was easily adapted in [15] (2009) to prove that a module alphabet R A has theEP for Hamming weight if and only if A is pseudo-injective with cyclic socle.On the other lane, in [15], J. Wood proved that Frobenius bimodules have the EP for swc, and in[9] it was shown that having a cyclic socle is sufficient (Theorem 3.4), while the necessity remainedan open question. Here, the necessity is proved, making use of a new notion, namely, the annihilatorweight , defined in section 4 below. 1 Background in Ring Theory
Let R be a finite ring with unity, denote by rad R its Jacobson radical, by the Wedderburn-Artintheorem (and Wedderburn’s little theorem) the ring R/ rad R is semi-simple, and (as rings) R/ rad R ∼ = k M i =1 M µ i ( F q i ) , (2.1)where each q i is a prime power, F q i denotes a finite field of order q i , and M µ i ( F q i ) denotes the ring of µ i × µ i matrices over F q i .It follows that, as left R -modules, R ( R/ rad R ) ∼ = k M i =1 µ i T i , (2.2)where R T i is the pullback to R of the matrix module M µi ( F qi ) M µ i × ( F q i ) via the isomorphism in equa-tion (2.1). It is known that these T i ’s form the complete list, up to isomorphism, of all simple left R -modules, hence the socle of any R -module A can be expressed as soc( A ) ∼ = k M i =1 s i T i , where s i is the number of copies of T i inside A .The next two propositions can be found in [15], page 17. Proposition 2.1. soc( A ) is cyclic if and only if s i ≤ µ i for i = 1 , . . . , k ; µ i defined as above. Proposition 2.2. soc( A ) is cyclic if and only if A can be embedded into R b R , the character group of R equipped with the standard module structure. The next theorem (Theorem 4.1, [14]), by J. Wood, was the key to carry out the strategy of Dinhand López-Permouth mentioned in the introduction, actually, it displays a thoughtfully constructedpiece-of-art example for the failure of the Hamming weight EP.
Theorem 2.3.
Let R = M m ( F q ) and A = M m × k ( F q ) . If k > m , there exist linear codes C + , C − ⊂ A N , N = k − Q i =1 (1 + q i ) , and an R -linear isomorphism f : C + → C − that preserves Hamming weight,yet there is no monomial transformation extending f . If soc( A ) is not cyclic, then the previous theorem, applied to a certain submodule of soc( A ) , givescounter-examples that pull back to give counter-examples for the original module, as the proof of thefollowing theorem shows (a detailed proof is found in [15], Theorem 6.4). Theorem 2.4. (Th. 5.2, [14]). Let R be a finite ring, and let A be a finite left R -module. If thereexists an index i and a multiplicity k > µ i so that kT i ⊂ soc( A ) ⊂ A , then the extension property forHamming weight fails for linear codes over the module A . Symmetrized Weight Compositions
Definition 3.1. (Symmetrized Weight Compositions) Let G be a subgroup of the automorphism group Aut R ( A ) of a finite R -module A . Define an equivalence relation ∼ on A : a ∼ b if a = bτ for some τ ∈ G . Let A/G denote the orbit space of this relation. The symmetrized weight composition (swc)built on G is a functionswc : A n × A/G → Q defined by, swc( x, a ) = |{ i : x i ∼ a }| , where x = ( x , . . . , x n ) ∈ A n and a ∈ A/G . Thus, swc counts the number of components in eachorbit.
Definition 3.2. (Monomial Transformation) Let G be a subgroup of Aut R ( A ) , a map T is called a G - monomial transformation of A n if there are some σ ∈ S n and τ i ∈ G for i = 1 , . . . , n , such that ( x , . . . , x n ) T = ( x σ (1) τ , . . . , x σ ( n ) τ n ) , where ( x , . . . , x n ) ∈ A n . Definition 3.3. (Extension Property) The alphabet A has the extension property (EP) with respect toswc if for every n , and any two linear codes C , C ⊂ A n , any R -linear isomorphism f : C → C preserving swc is extends to a G -monomial transformation of A n .In [12], J.A.Wood proved that Frobenius rings do have the extension property with respect to swc.Later, in [9], it was shown that, more generally, a left R -module A has the extension property withrespect to swc if it can be embedded in the character group b R (given the standard module structure). Theorem 3.4. (Th.4.1.3, [8]) Let A be a finite left R -module. If A can be embedded into b R (orequivalently, soc( A ) is cyclic), then A has the extension property with respect to the swc built on anysubgroup G of Aut R ( A ) . In particular, this theorem applies to Frobenius bimodules. We now define a new notion (the Midway!) on which we’ll depend in the rest of this paper.
Definition 4.1. (Annihilator Weight) On R A , define an equivalence relation ≈ by a ≈ b if Ann a =Ann b , where a and b are any two elements in A and Ann a = { r ∈ R | ra = 0 } is the annihilator of a .Clearly, Ann a is a left ideal.Now, on A n we can define the annihilator weight aw that counts the number of components in eachorbit. Remark:
It is easily seen that the EP for Hamming weight implies the EP for swc, and the EP for aw as well. Lemma 4.2.
Let R A be a pseudo-injective module. Then for any two elements a and b in A , a ≈ b ifand only if a ∼ b ( ∼ corresponds to the action of the whole group Aut R ( A ) ). roof. If a ∼ b , this means a = bτ for some τ ∈ Aut R ( A ) , and consequently Ann a = Ann b .Conversely, if a ≈ b , then we have (as left R -modules) Ra ∼ = R R/ Ann a = R R/ Ann b ∼ = Rb, with ra r + Ann a rb . By Proposition 5.1 in [15], since A is pseudo-injective, the isomorphism Ra → Rb ⊆ A extends to an automorphism of A taking a to b . Corollary 4.3. If R A is a pseudo-injective module, then the EP with respect to swc built on Aut R ( A ) is equivalent to the EP with respect to aw . Theorem 4.4.
Let R be a principal ideal ring, R A a pseudo-injective module, and let C be a submod-ule of A n for some n . Then a monomorphism f : C → A n preserves Hamming weight if and only if itpreserves swc built on Aut R ( A ) .Proof. The “if” part is direct. For the converse, we’ll use that any left ideal I contains an element e I that doesn’t belong to any other left ideal not containing I . Now, if ( c , c , . . . , c n ) f = ( b , b , . . . , b n ) , (4.1)choose, from c , c , . . . , c n ; b , b , . . . , b n , a component with a maximal annihilator I . Act on equation(4.1) by e I , then the only zero places are those of the components in equation (4.1) with annihilator I , and the preservation of Hamming weight gives the preservation of I -annihilated components. Omitthese components from the list c , c , . . . , c n ; b , b , . . . , b n and choose one with the new maximal,and repeat. This gives that f preserves aw and hence, by Lemma 4.2, f preserves swc built on Aut R ( A ) . Corollary 4.5. If R A is a module alphabet, then A has the extension property with respect to swc ifand only if soc( A ) is cyclic.Proof. The “if” part is answered by Theorem 3.4. Now, if soc( A ) is not cyclic, then by Proposition2.1, there is an index i such that s i > µ i , where s i T i ⊂ soc( A ) ⊂ A . Recall that T i is the pullbackto R of the matrix module M µi ( F qi ) M µ i × ( F q i ) , so that s i T i is the pullback to R of the M µ i ( F q i ) -module B = M µ i × s i ( F q i ) . Theorem 2.3 implies the existence of linear codes C + , C − ⊂ B N , and anisomorphism f : C + → C − that preserves Hamming weight, yet f does not extend to a monomialtransformation of B N . But the ring M µ i ( F q i ) is a principal ideal ring (in fact, more is true, Theoremix.3.7, [10]), besides, B is injective, and then Theorem 4.4 implies that f preserves swc built on Aut M µi ( F qi ) ( B ) .Now, a little notice finishes the work. The isomorphism in equation (2.1) and the projection map-pings R → R/ rad R → M µ i ( F q i ) allow us to consider the whole situation for C ± as R -modules. Since B pulls back to s i T i , we have C ± ⊂ ( s i T i ) N ⊂ soc( A ) N ⊂ A N , as R -modules. Thus C ± are linearcodes over A that are isomorphic through an isomorphism preserving swc built on Aut R ( s i T i ) . Also,any automorphism of A restricts to an automorphism of s i T i , hence the isomorphism preserves swcbuilt on Aut R ( A ) . However, this isomorphism does not extend to a monomial transformation of A N ,since, as appears in the proof of Theorem 2.3 (found in [14]), C + has an identically zero component,while C − does not. 4 eferences [1] H. Q. Dinh and S. R. López-Permouth, On the equivalence of codes over finite rings , Appl. Alge-bra Engrg. Comm. Comput. 15 (2004), no. 1, 37-50. MR MR2142429 (2006d:94097)[2] H. Q. Dinh and S. R. López-Permouth,
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