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Noetherian Semi-Perfect Rings of Distributive Module Type

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Noetherian Semi-Perfect Rings of Distributive Module Type ⥬ â¨ç­÷ âã¤÷ù. .8, ü1 Matematychni Studii. V.8, No.1 512.552.1 NOETHERIAN SEMI-PERFECT RINGS OF DISTRIBUTIVE MODULE TYPE Yu. Yaremenko Yu. Yaremenko. Noetherian semi-perfect rings of distributive module type, Matema- tychni Studii, 8(1997) 3{10. We prove that any noetherian semi-perfect ring of distributive module type is biserial. x1. Introduction. All rings considered in the paper are associative with 1 6= 0, modules are right and unitary. Further, noetherian (etc.) ring means two-side noetherian (etc.) ring. Recall that a module M is called distributive if K \ (L + N) = K \ L + K \ N for any submodules K, L, N. Clearly, submodules and quotient modules of a distributive module are distributive. A module is called semi-distributive if it is a direct sum of distributive modules. A ring is called right (left) semi-distributive if it is a right (left) semi-distributive module over itself. A right and left semi- distributive ring is called semi-distributive. Theorem 1.1. [1]. A module is distributive if and only if the socle of every its quotient module contains at most one copy of each simple module. Let R be the Jacobson radical of a ring A. A ring A is called semi-perfect if the quotient ring A=R is artinian and the idempotents can be lifted modulo R [2]. An idempotent e 2 A is called local if the ring eAe is local. Theorem 1.2. [3]. A ring A is semi-perfect if and only if the unity of A can be decomposed into a sum of mutually orthogonal local idempotents. Theorem 1.3. [4, x11.4], [5, x7]. A ring A is semi-perfect if and only if it decom- poses into direct sum of right ideals such that each one has exactly one maximal submodule. Denote by M n the direct sum of n copies of a module M, M 0 = 0. Therefore, n1 any semi-perfect ring A can be represented as the sum of right ideals A = P1 ⊕ · · · ⊕ ns , where are pairwise nonisomorphic modules and = , Ps P1;:::;Ps Ui Pi=PiR i = 1; : : : ; s, are simple. Modules P1;:::;Ps exhaust up to an isomorphism all indecomposable projective A-modules, while U1;:::;Us exhaust all nonisomorphic simple A-modules [6]. A semi-perfect right (left) semi-distributive ring is called an SP SDR-(SP SDL-) ring and semi-perfect semi-distributive ring is called an SP SD-ring. Theorem 1.4. [7] (see also [8, Theorem 4]). A semi-perfect ring A is an SP SDR- (SP SDL-)ring if and only if for any local idempotents e and f of the ring A the set eAf is a uniserial right fAf-module (uniserial left eAe-module). 1991 Mathematics Subject Classi cation. 16P40. Typeset by AMS-TEX 4 YU. YAREMENKO Corollary 1.5. [8]. Let A be a semi-perfect ring, 1 = e1 + ··· + en a decomposition of 1 2 A into the sum of mutually orthogonal local idempotents. The ring A is an SP SDR-(SP SDL-)ring if and only if for any idempotents ei and ej(i 6= j from the decomposition above the ring (ei +ej)A(ei +ej) is an SP SDR-(SP SDL-) ring. Corollary 1.6. [8]. Let A be a noetherian SP SD-ring, 1 = e1 + ··· + en a decom- position of the unity 1 2 A into the sum of mutually orthogonal local idempotents, Aij = eiAej and Ri the Jacobson radical of the ring Aii. Then RiAij = AijRj for any i; j = 1; : : : ; n. A module M over a ring A is called nitely-presented, if there exists an exact sequence P1 ! P0 ! M ! 0, where P1 and P0 are nitely-generated projective A-modules. De nition 1. [9]. A ring A is called a ring of distributive module type (DMT -ring) if every nitely presented A-module M is semi-distributive. Proposition 1.7. [9]. Let A be a semi-perfect DMT -ring, then the ring eAe is a DMT -ring for any nonzero idempotent e 2 A. An indecomposable module M is called biserial if it is distributive and contains chain submodules K1 and K2 (possibly zero) such that K1 + K2 is M or a maximal submodule in M and K1 \ K2 is trivial or simple [10]. An artinian ring A is called biserial if each left and right projective indecompos- able A-module is biserial [10]. De nition 2. [11]. A semi-perfect ring A is called biserial if every right and every left indecomposable projective A-module is biserial. Theorem 1.8. [12]. An artinian DMT -ring is biserial. The aim of the present paper is the proof of the following main theorem: Theorem I. A noetherian semi-perfect DMT -ring is biserial. De nition 3. [13]. A nite oriented graph Q (a quiver by Gabriel) is called biserial if each point of Q is a source of at most two arrows and each point of Q is a sink of at most two arrows, and Q does not contain double arrows. Let A be a semi-perfect ring such that the quiver Q(A) is well de ned (def. in [8], p.465). We de ne a quiver RQ(A) of A that is closely connected with the structure of A-modules [14], [15]. Let Q(A) contain n vertices 1; : : : ; n and assume that there exist tij arrows from i to j. We de ne RQ(A) as a quiver with vertices 1; : : : ; n; τ(1); : : : ; τ(n), where there exist tij arrows from i to τ(j). Thus RQ(A) is a biparted graph such that each arrow has a source in 1; : : : ; n and a sink in τ(1); : : : ; τ(n). For an oriented graph Q we will denote by Q a non-oriented graph with the same set of vertices and arrows as Q. Non-oriented graph of the type: 1 ! 2 ! 3 ! ···! m − 1 ! m is called a non-oriented chain. It is the Dynkin diagram Am. The following theorem is a corollary from the same results of [13, x5]. Theorem 1.9. Let A be a noetherian semi-perfect DMT -ring. Then A is an SP SD-ring with a biseriai quiver Q(A) and RQ (A) is a disjoint union of the Dynkin diagrams Am. Let R be the Jacobson radical of a semi-perfect ring A.The ring A is called reduced if the quotient ring A=R is a direct product of skew elds [6]. By the theorem of Morita (see, for example, [17]) the category of modules over an arbitrary semi-perfect ring is equivalent in a natural way to the category of modules over a reduced ring. Therefore, while studying semi-perfect DMT -rings NOETHERIAN SEMI-PERFECT RINGS OF DISTRIBUTIVE MODULE TYPE 5 one can consider only reduced rings. Indecomposable reduced noetherian semi- perfect DMT -rings will be called \good" rings (G-rings). x2. Minors of noetherian semi-perfect ring of a module distributive type. Following [16] we will call by minor of order n of a ring A a ring B of endomor- phisms of a nite generated projective A-module which can be decomposed into a direct sum of n indecomposable modules. From Proposition 1.7 one can obtain the following result. Proposition 2.1. Every minor of a noetherian semi-perfect DMT -ring is a noethe- rian semi-perfect DMT -ring. Proposition 2.2. [13 x5]. A local DMT -ring is uniserial. Corollary 2.3. [6]. A noetherian uniserial ring A is either a discrete valuation ring (may be non-commutative) or a uniserial Koethe ring, i.e. a uniserial artinian ring. We will describe reduced minors of the second and third order of noetherian semi-perfect DMT -rings. One can use the list of all up to isomorphism biregular quivers Q with two and three vertices such that RQ is a disjoint union of the Dynkin diagrams Am [13, x5]. Such quivers will be called admitted. Proposition 2.4. Admitted quivers Q for n = 2 and n = 3 can be de ned up to isomorphism by the following matrices. ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 1 0 1 1 1 0 1 1 1 = 2: ) ) ) ) ) ) n α 0 0 β 0 0 γ 0 1 δ 0 1 " 1 0 ζ 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 1 1 n = 3: 1) @ 0 0 1 A 2) @ 0 0 0 A 3) @ 0 0 1 A 4) @ 0 0 1 A 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 1 0 5) @ 0 0 1 A 6) @ 0 0 1 A 7) @ 0 1 1 A 8) @ 0 1 0 A 9) @ 0 0 1 A 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 0 10) @ 0 1 1 A 11) @ 0 1 1 A 12) @ 0 1 1 A 13) @ 0 1 0 A 14) @ 0 0 1 A 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 15) @ 0 1 1 A 16) @ 1 0 0 A 17) @ 1 0 1 A 18) @ 1 0 0 A 19) @ 1 1 0 A 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 20) @ 1 1 0 A 21) @ 0 0 1 A 22) @ 0 0 1 A 23) @ 0 0 1 A 24) @ 0 0 1 A 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 1 0 25) @ 0 0 1 A 26) @ 0 0 1 A 27) @ 1 0 1 A 28) @ 0 0 1 A 29) @ 0 0 1 A 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 30) @ 0 0 1 A 31) @ 0 0 1 A 32) @ 0 1 1 A : 1 1 0 1 1 0 1 1 0 Denote by Mn(R) the set of all real matrices of order n.
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