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On Semiperfect Rings of Injective Dimension One

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On Semiperfect Rings of Injective Dimension One S˜aoPaulo Journal of Mathematical Sciences 1, 1 (2007), 111–132 On semiperfect rings of injective dimension one V. V. Kirichenko Faculty of Mechanics and Mathematics, Kiev National Taras Shevchenko University, Vladimirskaya Str., 64, 01033 Kiev, Ukraine E-mail address: [email protected] Abstract. We give a characterization of right Noetherian semiprime semiperfect and semidistributive rings with inj. dimAAA 6 1. 1. Introduction Let Z be the ring of integers, Q be the field of rational numbers and p ∈ Z be a prime. Denote by Zp the following ring: nm o Z = ∈ Q | (n, p) = 1 . p n Obviously, every nonzero proper ideal J in Zp is principal and has the form k p Zp for some positive k. So, Zp is the principal ideal domain and all its ideals form the following descending chain: 2 k Zp ⊃ pZp ⊃ p Zp ⊃ ... ⊃ p Zp ⊃ .... ∞ T k Clearly, p Zp = 0. k=1 We have the following exact sequence 0 → Zp → Q → Q/Zp → 0. It is well-known that the Zp-modules Q and Q/Zp are injective (recall that ∞ Q/Zp is the abelian group p ). So, inj. dimZp Zp = 1. There are many papers devoted to study of injective dimension of rings (see, for example, [1], [2], [9], [3], [4]). 1991 Mathematics Subject Classification. 16P40; 16P20. Key words: injective dimension, exponent matrix, Gorenstein matrices, duality in tiled orders. 111 112 V. V. Kirichenko In the present paper we give a description of right Noetherian semiprime semiperfect and semidistributive rings with the injective dimension at most one. We will use the results and terminology of [11]. All rings are associative with nonzero identity. A ring A is decomposable if A = A1 × A2, otherwise A is indecomposable. Recall that a module M is called distributive if for all submodules K, L, N K ∩ (L + N) = K ∩ L + K ∩ N. Clearly, a submodule and a quotient module of a distributive module are distributive. A module is called semidistributive if it is a direct sum of distributive modules. A ring A is called right (left) semidistributive if the right (left) regular module AA (AA) is semidistributive. A right and left semidistributive ring is called semidistributive. Obviously, every uniserial module is distributive and every serial module is semidistributive. Theorem 1.1. [11, Theorem 14.1.6] A semiprimary right semidistributive ring A is right Artinian. Definition 1.2. The endomorphism ring of an indecomposable projective module over a semiperfect ring is called a principal endomorphism ring. The following is a decomposition theorem for semiprime semiperfect rings. Theorem 1.3. [11, Theorem 14.4.3] A semiprime semiperfect ring is a fi- nite direct product of indecomposable rings. An indecomposable semiprime semiperfect ring is either a simple Artinian ring or an indecomposable semiprime semiperfect ring such that all its principal endomorphism rings are non-Artinian. We write SP SD-ring A for a semiperfect and semidistributive ring A. Definition 1.4. A ring A is called semimaximal if it is a semiperfect semiprime right Noetherian ring such that for each local idempotent e ∈ A the ring eAe is a discrete valuation ring (not necessarily commutative), i.e., all principal endomorphism rings of A are discrete valuation rings. Proposition 1.5. [11, Proposition 14.4.12] A semimaximal ring is a finite direct product of prime semimaximal rings. Theorem 1.6. [11, Theorem 14.5.1] The following conditions for a semi- perfect semiprime right Noetherian ring A are equivalent: (a) A is semidistributive; S˜aoPaulo J.Math.Sci. 1, 1 (2007), 111–132 On semiperfect rings of injective dimension one 113 (b) A is a direct product of a semisimple Artinian ring and a semimax- imal ring. Theorem 1.7. [11, Theorem 14.5.2] Each semimaximal ring is isomorphic to a finite direct product of prime rings of the following form: 0 1 O πα12 O . πα1n O Bπα21 OO . πα2n OC A = B . C , (1) @ . .. A παn1 O παn2 O ... O where n > 1, O is a discrete valuation ring with a prime element π, and the αij are integers such that αij + αjk > αik for all i, j, k (αii = 0 for any i). A ring A is called a tiled order if it is a prime Noetherian SP SD-ring with nonzero Jacobson radical. Every tiled order is isomorphic to a ring of the form (1). Denote by Mn(B) the ring of all n × n-matrices over a ring B. Throughout of this paper, unless specifically noted, A denotes a tiled order with the classical ring of fractions Q = Mn(D), where D is the classical division ring of fractions of O. Definition 1.8. An integer matrix E = (αij) ∈ Mn(Z) is called • an exponent matrix if αij + αjk > αik and αii = 0 for all i, j, k; • a reduced exponent matrix if αij + αji > 0 for all i, j, i 6= j. We use the following notation: A = {O, E(A)}, where E(A) = (αij) is the exponent matrix of the ring A, i.e., Xn αij A = eijπ O, i,j=1 in which eij are the matrix units. If a tiled order is reduced, i.e. A/R(A) is the direct product of division rings, then αij + αji > 0 if i 6= j, i.e. E(A) is reduced. As usually, R(A) denotes the Jacobson radical of A. It is well-known that every semiperfect ring is Morita equivalent to a reduced semiperfect ring. Let M be a right A-module. Denote the injective dimension of M by inj. dim AM. Let AA (AA) be the right (left) regular A-module, i.e. AA is the right module over itself. Obviously, if A and B are Morita equivalent semiperfect rings, then inj. dimAAA = inj. dimBBB. Hence, every right Noetherian semiprime SP SD-ring A is Morita equ- ivalent to A1 × ... × Am, where Ai is either a division ring Di, or Ai = S˜aoPaulo J.Math.Sci. 1, 1 (2007), 111–132 114 V. V. Kirichenko {Oi, E(Ai)}, here E(Ai) is the reduced exponent matrix. Note that inj. dim Di Di equals zero. Definition 1.9. A reduced exponent matrix E = (αij) ∈ Mn(Z) is called Gorenstein if there exists a permutation σ of {1, 2, . , n} such that αik + αkσ(i) = αiσ(i) for i, k = 1, . , n. The main result of this paper is the following theorem Main Theorem. Let A = {O, E(A)} be a reduced prime Noetherian SP SD-ring with exponent matrix E(A) = (αij) ∈ Mn(Z). Then inj. dimAAA = 1 if and only if the matrix E(A) is Gorenstein. In this case inj.dimAA A = 1. 2. Tiled orders over discrete valuation rings and exponent matrices Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such orders and their quivers are com- pletely determined by its exponent matrices. Definition 2.1. Let A be a tiled order. A right (left) A-lattice is a right (left) A-module which is a finitely generated free O-module. In particular, all finitely generated projective A-modules are A-lattices. We shall denote by Latr(A) (resp. Latl(A)) the category of right (resp. left) A-lattices. Among all A-lattices we single out the so-called irreducible A-lattices, i.e., A-lattices contained in a simple right (resp. left) Q-module U (resp. V ). These lattices form a poset Sr(A) (resp. Sl(A)) with respect to inclu- sion. As it was shown in [11, Section 14.5], any right (resp. left) irreducible A-lattice M (resp. N) lying in U (resp. in V ) is an A-module with O-basis α αn (π 1 e1, . , π en), with ½ αi + αij > αj, if (α1, . , αn) ∈ Sr(Λ) T (2) αj + αij > αi, if (α1, . , αn) ∈ Sl(Λ), where T stands for the transposition operation. For our purposes, it suffices to consider a reduced tiled order A. In this case, the elements of Sr(A)(Sl(A)) are in a bijective correspondence T with the integer-valued row vectors ~a = (α1, . , αn) (column vectors ~a = T T (α1, . , αn) ), where ~a and ~a satisfy the conditions (2). We shall write [M] = (α1, . , αn) or M = (α1, . , αn), if M ∈ Sr(A). S˜aoPaulo J.Math.Sci. 1, 1 (2007), 111–132 On semiperfect rings of injective dimension one 115 Let ~b = (β1, . , βn). The order relation ~a ¹ ~b in Sr(A) is defined as follow: ~a ¹ ~b ⇐⇒ αi > βi for i = 1, . , n. Since A is a semidistributive ring, Sr(A) and Sl(A) are distributive lat- tices with respect to addition and intersection. Proposition 2.2. There exists only a finite number of irreducible A-lattices up to an isomorphism. Proof. Let A = {O, E(A)} be a tiled order with an exponent matrix E(A) = (αij). We can assume that αij > 0 for i, j = 1, . , n. Let M = (α1, . , αn) ∈ Sr(A). Considering an isomorphic module we can assume that all α1, ..., αn are positive integers. Denote a = min (α1, . , αn). Then M1 = (α1 − a, . , αn − a) is an irreducible A-lattice and M1 ' M. Suppose that αi = a. Then M1 = (β1, . , βn), where β1, . , βn are non-negative and βi = 0. Consequently, every irreducible A-lattice M is isomorphic to the lattice M1 with at least one zero coordinate. We obtain from (2) that 0 6 βj 6 αij. So, number of irreducible A-lattices of the form M1 is finite. ¤ Using the properties of projective covers of finitely generated modules over semiperfect rings, one can characterize projective modules of Sr(A) (resp.
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