In this paper, we study the structure of finite rings over which all free codes are splitting. In particular, we show that over the matrix rings over finite local rings all free codes are splitting

Hirano, Yasuyuki

Okayama University Scientific Achievement Repository

Math. J. Okayama Univ. 58 (2016), 133–140ON FINITE RINGS OVER WHICH ALL FREE CODESARE SPLITTINGYasuyuki HiranoAbstract. In this paper, we study the structure of finite rings overwhich all free codes are splitting. In particular, we show that over thematrix rings over finite local rings all free codes are splitting.1. IntroductionRecently linear codes over finite rings have raised a great interest for theirrole in algebraic coding theory. Firstly classical algebraic coding theory overfinite fields has been extended to algebraic coding theory over Galois ringsand finite commutative rings (see [3]). Next, algebraic coding theory overfinite noncommutative rings is studied by many authors (see the references in[6]). In particular, Wood [8] showed that linear coding theory is particularlywell-behaved over finite Frobenius rings (see also [9]). Finite Frobenius ringshave many good properties. Over a finite Frobenius ring, every projectivemodule is injective, and hence all free codes are splitting. In this paper, weassign light to only this property and we study the structure of finite ringswith this property: (†) All free codes are splitting. First we show that finiterings which are Morita equivalent to finite commutative rings have property(†). Next we give a necessary and sufficient condition for a free code C overany finite ring to be splitting. Then we give a sufficient condition for a finitering to have property (†). Using this result, we show that matrix rings overfinite local rings have property (†). Finally we give an example of a finiteindecomposable ring with property (†), but it is not quasi-Frobenius andalso is not Morita equivalent to any finite local ring.2. Preliminary resultsFirst we give some definitions. Let n be a positive number. FollowingGreferath [5], a linear (left) code C of length n over a finite ring R is asubmodule of RRn. We call C splitting if it is a direct summand of RRn. IfC has a basis, then C is said to be free. If C ∼= RRm for some m(≦ n), thenC is called a free code of rank m.For some notations and fundamental results on finite rings, we refer thereader to McDonald [7].Mathematics Subject Classification. Primary 16P10; Secondary 94B05.Key words and phrases. finite rings, ring-linear codes, free codes.133134 YASUYUKI HIRANOIn this paper, we consider the structure of finite rings R over which everyfree codes of length n over R is splitting for every natural number n. Inthis section, we state some general results on rings over which every finitelygenerated free submodule N of any finitely generated free module M is adirect summand of M .Proposition 2.1. The following conditions are equivalent for a ring R:(1) Every finitely generated free submodule N of any finitely generatedfree module M is a direct summand of M .(2) Every finitely generated projective submodule P of any finitely gener-ated projective module Q is a direct summand of Q.Proof. (1) ⇒ (2). Let P be a finitely generated projective submodule ofa finitely generated projective module Q. Then we find some projectivemodules P ′ and Q′ such that P ⊕ P ′ and Q ⊕ P ′ ⊕ Q′ are free modules.Since we assume that (1) holds, P ⊕P ′ is a direct summand of Q⊕P ′⊕Q′,that is Q⊕P ′⊕Q′ = P⊕P ′⊕X for some submoduleX of Q⊕P ′⊕Q′. Then,using modular law, we have Q = Q ∩ (Q⊕ P ′ ⊕Q′) = Q ∩ (P ⊕ P ′ ⊕X) =P ⊕ (Q ∩ P ′ ⊕Q′).(2) ⇒ (1). This is trivial. Proposition 2.2. Let R be a ring and consider the property:(‡) Every finitely generated free submodule N of any finitely generated freemodule M is a direct summand of M .Then there holds the following:(1) The property (‡) is Morita invariant.(2) Let e1, · · · , en are central and orthogonal idempotents of a ring Rsuch that e1 + · · ·+ en = 1. Then R has property (‡) if and only if all ringsRe1, · · · , Ren have property (‡).Proof. (1) Assume that two rings R and S are Morita equivalent via a cat-egory equivalence F : R-Mod → S-Mod and its inverse G : S-Mod → R-Mod. Assume that R has property (‡). Let 0 → N → M → M/N → 0be an exact sequence with projective modules N and M . Then, by [1,Propositions 21.4 and 21.6], 0 → G(N) → G(M) → G(M/N) → 0 be anexact sequence with projective modules G(N) and G(M). By hypothesis,the exact sequence 0 → G(N) → G(M) → G(M/N) → 0 splits. Then0 → FG(N) → FG(M) → FG(M/N) → 0 splits by [1, Propositions 21.4],This implies that 0 → N → M → M/N → 0 splits. By Proposition 2.1, Shas property (‡).(2) This follows from Proposition 2.1. ON FINITE RINGS OVER WHICH ALL FREE CODES ARE SPLITTING 1353. Finite commutative ringsA ring R with Jacobson radical J is called a local ring if R/J is a divisionring. Since a finite division ring is a field by Wedderburn’s theorem, a finitering R is local if R/J is a finite field.Lemma 3.1. Let R be a finite commutative local ring with Jacobson radicalJ . Let C be a free code of rank m of RRn. Then the code C¯ = (C +JRn)/JRn of R/J (R/J)n is a free code of rank m.Proof. We may assume that J 6= 0. We identify Rn/JRn with (R/J)n ina natural way. Then there is a positive integer k such that Jk 6= 0 andJk+1 = 0. Let C = Rx1⊕· · ·⊕Rxm be a free code of rank m with free basis{x1, · · · , xm} and let x¯i = xi+JRn ∈ (R/J)n for each i = 1, · · · ,m. Supposea¯1x¯1+ · · ·+ a¯mx¯m = 0 for some a¯i = ai+J ∈ R/J . Then a1x1+ · · ·+anxn ∈JRn. Since Jk 6= 0, we can take b1, · · · , bk ∈ J such that b1 · · · bk 6= 0. SinceJk+1 = 0, b1 · · · bka1x1 + · · · + b1 · · · bkamxm = 0. However x1, · · · , xm arelinearly independent over R, we obtain b1 · · · bka1 = · · · = b1 · · · bkam = 0.Since R is a local ring, these mean a1, · · · , am ∈ J , that is a¯i = ai + J = 0for all i ∈ {1, · · · ,m}. Theorem 3.2. Let R be a finite commutative ring. Then every free code Cof RRn is splitting for any positive integer n.Proof. First consider the case when R is a local ring. Let C = Rx1 ⊕· · · ⊕ Rxm (m < n) with free basis {x1, · · · , xm} and let {y1, · · · , yn} bethe standard basis of RRn. By Lemma 3.1, C¯ = R¯x¯1 ⊕ · · · ⊕ R¯x¯m is afree code of rank m of R¯y¯1 ⊕ · · · ⊕ R¯y¯n = (R/J)n. Then we can selecty¯i1 , · · · , y¯in−m from {y¯1, · · · , y¯n} such that {x¯1, · · · , x¯m, y¯i1 , · · · , y¯in−m} isa basis of (R/J)n. Then Rn = (Rx1 + · · ·+Rxn) + (Ryi1 + · · ·+Ryin−m) +JRn = Rn. Since JRn is small in Rn, we obtain Rn = (Rx1 + · · ·+Rxn) +(Ryi1 + · · · +Ryin−m) = Rn. This means that Rn is a homomorphic imageof (Rx1 ⊕ · · · ⊕ Rxn) ⊕ (Ryi1 ⊕ · · · ⊕ Ryin−m). Since Rn is a free moduleof rank n over a finite commutative local ring R, by Theorem of Jordan-Ho¨lder ([2, Theorerm 2.5.2]), the composition length of Rn is equal to thatof (Rx1 ⊕ · · · ⊕ Rxn) ⊕ (Ryi1 ⊕ · · · ⊕ Ryin−m). Therefore we conclude thatRn = (Rx1⊕· · ·⊕Rxn)⊕ (Ryi1⊕· · ·⊕Ryin−m) = C⊕ (Ryi1⊕· · ·⊕Ryin−m).Next consider the case when R is a finite commutative ring. We caneasily see that R is a direct sum of finite commutative local rings. So letR = Re1 ⊕ · · · ⊕ Rek, where each Rei is a local ring. Now let C be a freecode of RRn. Then, for each i, eiC be a free Rei-submodule of rank m of afree Rei-module eiRn. Then eiRn = eiC⊕C′i for some Rei-submodule C′i ofeiRn. Then Rn = (e1C⊕· · ·⊕ekC)⊕(C′1⊕· · ·⊕C′k) = C⊕(C′1⊕· · ·⊕C′k). 136 YASUYUKI HIRANO4. Non-commutative artinian ringsWe denote the ring of n×n-matrices over a ringR by Mn(R). By Theorem3.2 and Proposition 2.2 we have the following proposition.Proposition 4.1. Let R1, · · · , Rk be finite commutative rings. Then, forany positive integers n1, · · · , nk, the ring R = Mn1(R1)⊕ · · · ⊕Mnk(Rk) hasthe following property:(†) Every free code C of RRn is splitting for every positive integer n.However the following example shows that a non-commutative finite ringneed not have property (†).Example 1. LetK be a finite field and consider the subringR =(K K0 K)of the ring(K KK K)of 2×2 matrices overK. Since the left ideal(0 K0 0)of R is isomorphic to the left ideal(K 00 0), the code C =(0 K0 0)⊕(0 K0 K)of R⊕R = R2 is isomorphic to RR, and hence C is a free code.However C is not a direct summand of R2, because(0 K0 0)is small inRR.Next we give a necessary and sufficient condition for a free code C over afinite ring to be splitting.Theorem 4.2. Let R be a finite ring and let C be a free code of RRn. Thenthe following are equivalent:(1) C is splitting.(2) C¯ = (C + JRn)/JRn is a free code of R/J (R/J)n and the rank of C¯equals the rank of C.Proof. (1) ⇒ (2). This is clear.(2) ⇒ (1). Assume the rank of C is m. Since R/J is a finite semisimplering, (R/J)n = (C+JRn)/JRn⊕L for some code L of (R/J)n. Clearly L isa free code of rank n−m. Hence there exist x1, · · · , xn−m ∈ RRn such that{x¯1, · · · , x¯n−m} is a free basis of L over R/J . Then Rn = C + (Rx1 + · · ·+Rxn−m) + JRn. By Nakayama’s lemma, we obtain Rn = C + (Rx1 + · · ·+Rxn−m). Since C is of rankm over R, there is an isomorphism f : RRm → C.Also there is an epimorphism g : RRn−m → Rx1+· · ·+Rxn−m. Since RRn =RRm ⊕ RRn−m, we can define an epimorphism φ : RRn → C + (Rx1 + · · ·+Rxn−m) = RRn by φ = f on RRm and φ = g on RRn−m. Since RRn is a finiteset, the epimorphism φ is an isomorphism. Therefore Rx1 + · · ·+Rxn−m isa free R-module of rank n−m and Rn = C ⊕ (Rx1 + · · ·+Rxn−m). ON FINITE RINGS OVER WHICH ALL FREE CODES ARE SPLITTING 137In the proof of Theorem 4.2, we used that fact that RRn is a finite set.But we can also use the following easy lemma.Lemma 4.3. Let R be a ring and let M be a left R-module of finite com-position length. Consider an R-endomorphism f of M . Then the followingare equivalent:(1) f is an epimorphism.(2) f is a monomorphism.(3) f is an isomorphism.Using the lemma above, we can generalize Theorem 4.2 to left artinianrings.Theorem 4.4. Let R be a left artinian ring and let N be a finitely generatedfree submodule of a finitely generated free module M . Then the following areequivalent:(1) N is a direct summand of M .(2) N¯ = (N + JM)/JM is a free R/J-module and the rank of N¯ overR/J equals the rank of N over R.A left principal indecomposable module of a ring R is a left submodule ofRR that is a direct summand of R and is an indecomposable module. Nowwe state a sufficient condition for a left artinian ring to have the propertythat every finitely generated free submodule N of any finitely generated freemodule M is a direct summand of M .Theorem 4.5. Let R be a left artinian ring with non-zero Jacobson radical Jof nilpotency index k+1. Assume that for every left principal indecomposablemodule L, JkL 6= 0. Then every finitely generated free submodule N of anyfinitely generated free module M is a direct summand of M .Proof. Since R is a left artinian ring, by [2, Proposition 2.5.6] N is a directsum of indecomposable modules, say N = L1⊕ · · · ⊕Ln. Since N ∼= Rm forsome positive integer m, by Theorem of Krull-Schmidt ([2, Theorem 2.5.11])each Li is isomorphic some indecomposable direct summand of RR.Let us set L¯i = (Li+JM)/JM ⊂M/JM . We claim that L¯i is a simple R-module for all i ∈ {1, · · · , n}. If L¯i = 0 for some i, then Li ⊂ JM , and henceJkLi ⊂ Jk+1M = 0. This contradicts our assumption. Therefore L¯i 6= 0for all i ∈ {1, · · · , n}. There is an isomorphism L¯i = (Li + JM)/JM ∼=Li/(Li∩JM). Since JLi ⊂ Li∩JM and since Li/JLi is a simple R-moduleby [1, Proposition 27.10], we conclude that L¯i is a simple R-module for alli ∈ {1, · · · , n}.Next we claim that L¯1+ · · ·+ L¯n is a direct sum. Assume a¯1+ · · ·+ a¯n =0 ∈ M/JM , where ai ∈ Li and a¯i = ai + JM for each i = 1, · · · , n. Thena1 + · · · + an ∈ JM and so Jk(a1 + · · · + an) = 0. Since L1 + · · · + Ln is a138 YASUYUKI HIRANOdirect sum, we obtain Jkai = 0 for each i = 1, · · · , n. By our assumption,JkLi 6= 0 and Li/JLi is simple, we conclude that ai ∈ JLi ⊂ JM for eachi = 1, · · · , n. Hence a¯1 = · · · = a¯n = 0. This proves our claim.Hence N¯ = L¯1 ⊕ · · · ⊕ L¯n is isomorphic to R/J (R/J)m. Thus the rank ofN¯ over R/J and the rank of N over R are same m. Therefore by Theorem4.4 N is a direct summand of M . Remark. If R be a left artinian ring (or more generally, if R is a semiperfectring), then there are finitely many left principal indecomposable modulesL1, · · · , Lm such that every finitely generated indecomposable projectiveleft R-module is isomorphic to one and only one of L1, · · · , Lm (see [1,Proposition 27.10]). Hence we need only check the condition of Theorem4.5 holds for L1, · · · , Lm.We can easily see that local rings satisfy the condition in Theorem 4.5.Hence we have the following corollary.Corollary 4.6. Let R be a finite local ring. Every free code C of RRn issplitting for any positive integer n.By Corollry 4.6 and Proposition 2.2 we obtain the following theorem.Theorem 4.7. Let R1, · · · , Rk be finite local rings. Then, for any positiveintegers n1, · · · , nk, the ring R = Mn1(R1)⊕· · ·⊕Mnk(Rk) has the followingproperty:(†) Every free code C of RRn is splitting for any positive integer n.Finally we give an example of a finite indecomposable ring with property(†), but it is not quasi-Frobenius and also is not Morita equivalent to anyfinite local ring.Example 2. Let K be a finite field and let A = {(a b0 a)| a ∈ K, b ∈K ⊕K}.Consider the following subring of M2(A):R ={a 0 0 b0 a 0 00 c d 00 0 0 d | a, d ∈ K, b, c ∈ K ⊕K}. Then we see J(R) ={0 0 0 b0 0 0 00 c 0 00 0 0 0 | b, c ∈ K⊕K}and R/J(R) ∼= K⊕K. We can also seeJ(R)2 = 0, and hence J(R) is nilpotent of index 2. Consider the idempotentON FINITE RINGS OVER WHICH ALL FREE CODES ARE SPLITTING 139e =1 0 0 00 1 0 00 0 0 00 0 0 0. Then L1 = Re is an indecomposable left ideal of Rand the socle of L1 is S1 ={0 0 0 00 0 0 00 c 0 00 0 0 0 | c ∈ K⊕K}. We can easilysee that JL1 = S1 6= 0. If we set L2 = R(1 − e), then L2 is also an inde-composable left ideal of R. The socle of L2 is S2 ={0 0 0 b0 0 0 00 0 0 00 0 0 0 | b ∈K ⊕ K}and we can see that JL2 = S2 6= 0. By Theorem 4.5, every freecode C of RRn is splitting for every positive integer n. Clearly S1 and S2are not simple. Hence R is not quasi-Frobenius (cf. [4, Theorem 9.3.7]).Also R is not Morita equivalent to any finite local ring.References[1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Second Edition,Springer-Verlag, New York-Heidelberg-Berlin, 1992.[2] J. Beachy, Introductory Lectures on Rings and Modules, London Mathematical SocietyStudent Texts, 47, Cambridge University Press, Cambridge, 1999.[3] G. Bini and F. Flamini, Finite Commutative Rings and Their Applications, KluwerAcademic Press Publishers, 2002.[4] Y. A. Drozd and V. V. Kirichenko, Finite Dimensional Algebras, Springer-Verlag,Berlin, 1994.[5] M. Greferath, Cyclic codes over finite rings, Discrete Math., 177, no. 1 (1997), 273–277.[6] M. Greferath, An Introduction to Ring-Linear Coding Theory, Gro¨bner Bases, Coding,and Cryptography, Edited by M. Sala, T. M. L. Perret, S. Sakata and C.Traverso,Springer-Verlag, Berlin, 2009, 219–238.[7] B. R. McDonald, Finite Rings With Identity, (Pure and Applied Mathematics, Vol.28), Marcel Dekker, Inc., New York, 1974.[8] J. A. Wood, Duality for modules over finite rings and applications to coding theory,Amer. J. Math., 121, no. 3 (1999), 555-575.[9] J. A. Wood, Applications of finite Frobenius rings to the foundations of algebraic cod-ing theory, Proceedings of the 44th Symposium on Ring Theory and RepresentationTheory, Nagoya, 2012, 223–245.140 YASUYUKI HIRANOYasuyuki HiranoDepartment of MathematicsNaruto university of EducationNaruto, 772-8502 Japane-mail address: yahirano@naruto-u.ac.jp(Received September 11, 2013 )(Accepted October 8, 2013 )

On finite rings over which every free codes is splitting

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