Serial piecewise domains

Serial piecewise domains

A ring A is called a piecewise domain with respect to the complete set of idempotents {e1,e2,…,em} if every nonzero homomorphism eiA→ejA is a monomorphism. In this paper we study the rings for which conditions of being piecewise domain and being hereditary (or semihereditary) rings are equivale...

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Дата:2007
Автори: Gubareni, N., Khibina, M.
Формат: Стаття
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Опубліковано: Інститут прикладної математики і механіки НАН України 2007
Назва видання:Algebra and Discrete Mathematics
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Цитувати:Serial piecewise domains / N. Gubareni, M. Khibina // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 59–72. — Бібліогр.: 25 назв. — англ.

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spelling irk-123456789-1523822019-06-11T01:25:28Z Serial piecewise domains Gubareni, N. Khibina, M. A ring A is called a piecewise domain with respect to the complete set of idempotents {e1,e2,…,em} if every nonzero homomorphism eiA→ejA is a monomorphism. In this paper we study the rings for which conditions of being piecewise domain and being hereditary (or semihereditary) rings are equivalent. We prove that a serial right Noetherian ring is a piecewise domain if and only if it is right hereditary. And we prove that a serial ring with right Noetherian diagonal is a piecewise domain if and only if it is semihereditary. 2007 Article Serial piecewise domains / N. Gubareni, M. Khibina // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 59–72. — Бібліогр.: 25 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16P40, 16G10 http://dspace.nbuv.gov.ua/handle/123456789/152382 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description A ring A is called a piecewise domain with respect to the complete set of idempotents {e1,e2,…,em} if every nonzero homomorphism eiA→ejA is a monomorphism. In this paper we study the rings for which conditions of being piecewise domain and being hereditary (or semihereditary) rings are equivalent. We prove that a serial right Noetherian ring is a piecewise domain if and only if it is right hereditary. And we prove that a serial ring with right Noetherian diagonal is a piecewise domain if and only if it is semihereditary.
format Article
author Gubareni, N.
Khibina, M.
spellingShingle Gubareni, N.
Khibina, M.
Serial piecewise domains
Algebra and Discrete Mathematics
author_facet Gubareni, N.
Khibina, M.
author_sort Gubareni, N.
title Serial piecewise domains
title_short Serial piecewise domains
title_full Serial piecewise domains
title_fullStr Serial piecewise domains
title_full_unstemmed Serial piecewise domains
title_sort serial piecewise domains
publisher Інститут прикладної математики і механіки НАН України
publishDate 2007
url http://dspace.nbuv.gov.ua/handle/123456789/152382
citation_txt Serial piecewise domains / N. Gubareni, M. Khibina // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 59–72. — Бібліогр.: 25 назв. — англ.
series Algebra and Discrete Mathematics
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2007). pp. 59 – 72 c© Journal “Algebra and Discrete Mathematics” Serial piecewise domains Nadiya Gubareni and Marina Khibina Communicated by M. Ya. Komarnytskyj Dedicated to Professor V. V. Kirichenko on the occasion of his 65th birthday Abstract. A ring A is called a piecewise domain with respect to the complete set of idempotents {e1, e2, . . . , em} if every nonzero homomorphism eiA → ejA is a monomorphism. In this paper we study the rings for which conditions of being piecewise domain and being hereditary (or semihereditary) rings are equivalent. We prove that a serial right Noetherian ring is a piecewise domain if and only if it is right hereditary. And we prove that a serial ring with right Noetherian diagonal is a piecewise domain if and only if it is semihereditary. 1. Introduction All rings considered in this paper are assumed to be associative with iden- tity 1 6= 0, and all modules are unitary right modules, unless otherwise specified. This paper is devoted to considering the structure of some classes of piecewise domains. These rings first were introduced and studied by R.Gordon and L.W.Small [11]. Piecewise domains extend the notion of hereditary and semihereditary rings. Recall that a ring A is a right hereditary (resp. semihereditary) if any its ideal (resp. finitely generated ideal) is projective. Any princi- pal ideal domain is hereditary. Any Dedekind ring is hereditary and any 2000 Mathematics Subject Classification: 16P40, 16G10. Key words and phrases: piecewise domain, hereditary ring, semihereditary ring, serial ring, Noetherian diagonal, prime radical, prime quiver. Jo u rn al A lg eb ra D is cr et e M at h .60 Serial piecewise domains Prüfer ring is semihereditary. Really, the property being hereditary (resp. semihereditary) is invariant with respect to Morita equivalence. In par- ticular, any full matrix ring over a Dedekind domain (resp. Prüfer ring) is hereditary (semihereditary). It is true the following statement. Theorem 1.1. ([3]). A ring A is right hereditary (resp. semihereditary) if and only if any (finitely generated) submodule of a right projective A- module is projective. Example 1.1. Let A = ( Z Q 0 Z ) , where Z is the ring of integers and Q is the field of rational numbers. Then A is a two-sided Noetherian piecewise domain. A is a semihereditary ring, but it is not hereditary. Example 1.2. ([18], p.46, Small’s example). Let A = ( Z Q 0 Q ) , where Z is the ring of integers and Q is the field of rational numbers. Then A is a semihereditary, and so it is piecewise domain, it is right hereditary but it is not left hereditary. The following examples show that there are some very wide class of rings which are piecewise domains but not hereditary rings. Example 1.3. Let T2(Z) = ( Z Z 0 Z ) , where Z is the ring of integers and Q is the field of rational numbers. Then T2(Z) is a two-sided Noetherian piecewise domain, but it is not hereditary. Example 1.4. Let P = {p1, p2, . . . , pn} be a partial ordered connected set with partial order �, and let A = Mn(Z) be a full ring of all square matrices of order n, where Z is the ring of integers. Denote by eij the matrix units of A (i, j = 1, . . . , n). Let Bn(Z,P) be a subring of A which is consisted from the matrices of the following form: B = (bij) ∈ A, where bij ∈ eiiAejj , and bij = 0 if pi � pj . The ring Bn(Z,P) is a piecewise domain, but it is not a hereditary ring for n ≥ 2, since there exists an idempotent e = eii + ejj (for i 6= j) such that the ring eBn(Z,P)e ≃ T2(Z) is not hereditary. Jo u rn al A lg eb ra D is cr et e M at h .N. Gubareni, M. Khibina 61 In this paper we study the rings for which conditions of being piece- wise domain and being hereditary (or semihereditary) rings are equiva- lent. The central structure theorem which was proved by R.Gordon and L.W.Small states that all piecewise domains are triangular in the sense of definition which will be given below. Definition 1.1. ([9], p.56). A ring A is called an FDI-ring if there exists a decomposition of the identity of 1 ∈ A into a finite sum 1 = e1 + e2 + . . . + en of pairwise orthogonal primitive idempotents ei. A right projective A-module P of an FDI-ring A is called principal if P ≃ eiA for i = 1, . . . , n. The important examples of FDI-rings are semiperfect, Artinian, Noe- therian, Goldie rings. Note that the decomposition of 1 ∈ A, giving in the definition of an FDI-ring, may be non-unique. Definition 1.2. ([1], p.89). A finite orthogonal set of idempotents e1, e2, . . ., em ∈ A is called complete if e1 + e2 + . . . + em = 1 ∈ A Definition 1.3. ([11]). A ring A is said to have enough idempotents if it has a complete set of orthogonal primitive idempotents. Remark 1.1. Note that the concept of a ring to have enough idempotents coincides with concept of an FDI-ring. 2. Piecewise domains In this section we consider some properties of piecewise domains. Definition 2.1. A ring A is called a piecewise domain with respect to a complete set of idempotents {e1, e2, . . . , em} if every nonzero homomor- phism eiA → ejA is a monomorphism. Piecewise domain were first introduced and studied by R.Gordon and L.Small in the paper [11]. Since for a piecewise domain it follows that eiAei is a domain for any i = 1, . . . , m, the set of {e1, e2, . . . , em} is a complete set of primitive orthogonal idempotents, and so any piecewise domain A is an FDI-ring. The important examples of piecewise domains are hereditary and semihereditary FDI-rings: Jo u rn al A lg eb ra D is cr et e M at h .62 Serial piecewise domains Proposition 2.1. ([8], proposition 10.7.9). Any right semihereditary FDI-ring is a piecewise domain. The following structure theorem describes piecewise domains in the general case. Theorem 2.1. ( [11], main theorem). If A is a piecewise domain, then A =          B1 B12 . . . . . . B1r 0 . . . . . . ... ... . . . . . . . . . ... ... . . . . . . Br−1r 0 . . . . . . 0 Br          , where each Bij is a Bi-Bj-bimodule and each Bi is a prime piecewise domain of the form B =       O1 . . . . . . O1t ... O2 ... ... . . . ... Ot1 . . . . . . Ot       , where each Oj is a domain and each Ojk is isomorphic as a right Ok- module to a nonzero right ideal in Ok and as a left Oj-module to a nonzero left ideal in Oj. Moreover, the integer r is uniquely determined by A. Corollary 2.1. The prime radical Pr(A) = N of a piecewise domain A is of the form N = Pr(A) =                   0 B12 B13 . . . . . . B1r 0 0 B23 . . . B2r ... . . . . . . . . . . . . ... ... . . . . . . Br−2r−1 Br−2r ... . . . . . . Br−1r 0 . . . . . . . . . 0 0                   , and so N is nilpotent. Jo u rn al A lg eb ra D is cr et e M at h .N. Gubareni, M. Khibina 63 Definition 2.2. A decomposition of identity 1 = f1 + f2 + · · ·+ fm of a ring A, where fi are idempotents, is called triangular if fiAfj = 0 for all i > j. Such a decomposition is called prime if fiAfi is a prime ring for any i = 1, . . . , m. A ring A is called triangular if there exists a triangular decomposition of the identity of A. A ring A is said to be a primely triangular ring if there exists a tri- angular prime decomposition of the identity of A, i.e., there exists a de- composition of the identity 1 ∈ A into a finite sum 1 = f1 + f2 + · · ·+ fm of pairwise orthogonal idempotents such that fiAfj = 0 for all i > j and fiAfi is a prime ring for any i = 1, . . . , m. Remark 2.1. Note that the termin "a triangular ring" was first in- troduced by S.U.Chase in 1961 [2] for a semiprimary ring where all Ri = eiAei are simple Artinian rings. This termin was used by L.W.Small for arbitrary Noetherian rings [23]. M.Harada in 1966 introduced the ter- min "generalized triangular matrix rings" for rings with triangular decom- position of the identity where Ri = eiAei are arbitrary rings [10]. And he studied the properties of such rings when Ri are semiprimary rings. It is obvious that in this case the generalized triangular matrix rings are also semiprimary. Yu.A.Drozd in 1980 used the termin "a triangular ring" for rings with triangular prime decomposition of the identity [6]. Corollary 2.2. A piecewise domain is a primely triangular ring. In particular, any semihereditary FDI-ring and any hereditary FDI-ring is a primely triangular ring. Definition 2.3. ([8], p.291) Let Pr(A) be the prime radical of a ring A. The quotient ring A = A/Pr(A) is called the diagonal of A. A ring A is called an FDD-ring if A is an FD-ring. From theorem 2.1, corollary 2.1 and corollary 2.2 it immediately follow the following corollaries. Corollary 2.3. The diagonal A = A/Pr(A) of a piecewise domain A is an finite direct product of prime piecewise domains A = B1×B2×. . .×Br. Therefore A is an FD-ring, and A is an FDD-ring. Corollary 2.4. A piecewise domain A considered as a group is a direct sum of the ring A0 ≃ A and N = Pr(A): A = A0 ⊕ N . Since, by corollary 2.3, any piecewise domain A is an FDD-ring, we can build the prime quiver PQ(A) of A (see [8], section 11.7). Since the prime radical of a piecewise domain A is nilpotent, and so T -nilpotent, from theorem 11.7.3, [8], we immediately obtain the following statement. Jo u rn al A lg eb ra D is cr et e M at h .64 Serial piecewise domains Proposition 2.2. Let A be a piecewise domain. Then the prime quiver PQ(A) is connected if and only if the ring A is indecomposable. Recall that a quiver without multiple arrows and multiple loops is called simply laced. Proposition 2.3. Let A be a piecewise domain. Then the prime quiver PQ(A) is an acyclic simply laced quiver. Theorem 2.2. A right perfect piecewise domain is a semiprimary ring. Proof. Let A be a right perfect piecewise domain. Since any one-sided perfect ring is semiperfect, A is semiperfect. Therefore the prime rad- ical of A is nilpotent, by ([9], corollary 4.9.3). Since the prime radical of a one-sided perfect ring coincides with Jacobson radical R of A, by ([9], proposition 4.7.5), R is nilpotent. Thus A/R is Artinian and R is nilpotent, i.e., A is semiprimary. Since any right hereditary FDI-ring is a piecewise domain, by proposi- tion 2.1, we obtain as an immediately corollary the following result which was proved by M.Teply. Theorem 2.3. ([24]). A right perfect right hereditary ring is semipri- mary. 3. Serial right Noetherian piecewise domains Definition 3.1. ([8], p.300) A right A-module is called serial if it is de- composed into a direct sum of uniserial modules, that is, modules possess- ing a linear lattice of submodules. A ring which is right serial module and left serial module over itself is called a serial ring. Rings, over which all modules are serial, were first introduced and studied by G.Köthe [17] and T.Nakayama [20], [21]. Serial Artinian rings was studied by L.A.Skorniakov [22], K.R.Fuller [7], D.Eisenbud, P.Griffith [4], [5], G.Ivanov [13], and others. It was proved that a ring A is serial right Artinian ring if and only if A is a direct sum of uniserial right modules. Serial non-Artinian rings were first studied and described by V.V.Ki- richenko [14], [15] and R.B.Warfield [25]. In particular, they described the structure of serial Noetherian rings. Theorem 3.1. ([14]). Any serial Noetherian ring can be decomposed into a finite direct product of an Artinian serial ring and a number of semiper- fect Noetherian prime hereditary rings. Conversely, all such rings are serial and Noetherian. Jo u rn al A lg eb ra D is cr et e M at h .N. Gubareni, M. Khibina 65 The structure of semiperfect Noetherian prime hereditary rings was studied by G.Michler, who proved the following main theorem. Theorem 3.2. ([19]). A Noetherian semiperfect prime reduced hereditary ring A is either a division ring or it is isomorphic to a ring of the form Hm(O) =      O O . . . O M O . . . O ... ... . . . ... M M . . . O      , where O is a discrete valuation ring, and M is its unique maximal ideal. The full description of all serial right Noetherian rings are given by the following theorem. Theorem 3.3. ([14], [15]). Any serial right Noetherian ring is Morita equivalent to a direct product of a finite number of rings of the following types: 1) Artinian serial rings; 2) rings isomorphic to rings of the form Hm(O); 3) rings isomorphic to quotient rings of a ring H(O, m, n) = ( Hm(O) Mm,n(D) 0 Tn(D) ) , where O is a discrete valuation ring with skew field of fractions D, Tn(D) is the ring of upper triangular matrices of order n, and Mm,n(D) is a set of all rectangular matrices of size m×n over the division ring. Conversely, all rings of this form are serial and right Noetherian. The following theorem gives a description of serial semiprime and right Noetherian ring. Theorem 3.4. ([8], theorem 13.5.3). A serial semiprime and right Noe- therian ring can be decomposed into a direct product of prime rings. A serial prime and right Noetherian ring is also left Noetherian and two- sided hereditary. In the Artinian case such a ring is Morita equivalent to a division ring and in the non-Artinian case it is Morita equivalent to a ring isomorphic to Hm(O), where O is a discrete valuation ring. Conversely, all such rings are prime two-sided hereditary and Noetherian. From this theorem and corollary 2.3 we immediately obtain the fol- lowing statement. Jo u rn al A lg eb ra D is cr et e M at h .66 Serial piecewise domains Proposition 3.1. If A is a serial right Noetherian piecewise domain, then its diagonal A is Morita equivalent to the finite direct product of rings of the form D, where D is a division ring, or Hn(O), where O is a discrete valuation ring. Proposition 3.2. An Artinian serial piecewise domain is Morita equiv- alent to a direct product of rings of the form Tn(D), where D is a division ring. And so A is a hereditary ring. Proof. We can assume that a ring A is indecomposable. According to [8], theorems 11.1.9 and 12.1.2, one can assume that the quiver of A is a chain or a simple cycle. Consider the first case when the quiver Q(A) of A is a chain, then A is Morita equivalent to a ring isomorphic to Tn(D)/I, where I is a two-sided ideal of Tn(D) (see [8], pp. 306-308). Let AA = P1 ⊕ . . . ⊕ Pn, where Pi = eiiA, i = 1, . . . , n. Obviously, any nonzero two-sided ideal I ⊂ Tn(D) contains e11Tn(D)enn. So, if A 6= Tn(D) then any homomorphism ϕ : Pn → P1 is zero, i.e., e11Aenn = 0. The matrix units e12, e23, . . . , en−1n belong A because there is an ar- row i → i + 1 for each i = 1, . . . , n − 1 in the quiver Q(A). Therefore the product e12e23 . . . en−1n = e1n is nonzero, by [8], proposition 10.7.8, and e1n ∈ A. Thus e11Aenn 6= 0. We obtain a contradiction and so A = Tn(D). Consider the second case when the quiver Q(A) is a cycle: 1 2 n 1 • // • // . . . // • // • (3.1) Since A is right Artinian piecewise domain, by [8], proposition 11.2.3, the prime radical of A coincides with its Jacobson radical. So, in this case the prime quiver PQ(A) is obtained from the quiver Q(A) by changing all arrows going from one vertex to other vertex by one arrow. In our case Q(A) is a simple cycle (3.1). Thus Q(A) = PQ(A). By proposition 2.3, the prime quiver PQ(A) of a piecewise domain is an acyclic simply laced quiver. So we obtain a contradiction. Theorem 3.5. For a serial right Noetherian ring A the following condi- tions are equivalent: (i) A is right hereditary; (ii) A is a piecewise domain. Proof. (i) ⇒ (ii) follows from proposition 2.1. (ii) ⇒(i). By [8], theorem 13.4.3, any serial right Noetherian ring A is Morita equivalent to a direct sum of a finite number of rings of the following types: Jo u rn al A lg eb ra D is cr et e M at h .N. Gubareni, M. Khibina 67 (1) Artinian serial rings; (2) rings isomorphic to rings of the form Hn(O); (3) rings isomorphic to quotient rings of H(O, m, n), where O is a discrete valuation ring. If we have the case (1), then A is hereditary, by proposition 3.2. So consider case (2). An indecomposable reduced serial ring A is isomorphic to Hm(O), which is hereditary. Therefore all piecewise serial ring of the case (2) are hereditary. Case (3). We can assume that a ring A is reduced. By theorem 1.2.4, proposition 1.2.14 and theorem 11.7.3, [8], the prime quiver of an indecomposable serial piecewise domain A is a chain. So, there exists a decomposition of 1 ∈ A into a sum of mutually orthogonal idempotents 1 = f1 + . . .+ ft such that fiAfj = 0 for i > j, all rings Aii = Ai = fiAfi are prime and fiAfi+1 6= 0 for i = 1, . . . , t− 1. Consequently, Ai is either a division ring or a ring Hm(O) and A =              A1 A12 ∗ A2 A23 . . . . . . 0 At−1 At−1t At              (3.2) Since A is an indecomposable serial reduced piecewise domain of the type (3), A1 = Hm(O), A2 = D, . . . , At−1t = D. Consequently, matrix units e1,m+1, e2,m+1, . . . , em,m+1, em+1,m+2, . . ., em+t−2,m+t−1 lye in A. Therefore, all Peirce components fiAfj for i ≤ j are nonzero by [8], proposition 10.7.8, and A = H(O, m, t− 1). Thus, all piecewise domains in the case (3) are right hereditary. The theorem is proved. Corollary 3.1. For a Noetherian serial ring A the following conditions are equivalent: (1) A is two-sided hereditary; (2) A is a piecewise domain. Recall that a module M is called distributive if for all submodules K, L, N K ∩ (L + N) = K ∩ L + K ∩ N. A module is called semidistributive if it is a direct sum of distributive modules. A ring A is called right (resp. left) semidistributive if the right Jo u rn al A lg eb ra D is cr et e M at h .68 Serial piecewise domains (resp. left) regular module AA (AA) is semidistributive. A right and left semidistributive ring is called semidistributive (see [8], p.341). We write an SPSD-ring for a semiperfect semidistributive ring. The next examples show that the conditions of theorem 3.5 are not equivalent in the case if we change the property being serial by the prop- erty being an SPSD-ring neither in the case of a two-sided Artinian ring no in the case of a two-sided Noetherian ring. Example 3.1. Let A =     D D D D 0 D 0 D 0 0 D D 0 0 0 D     , (3.3) where D is a division ring. A is obviously an Artinian ring. Since for for any primitive orthog- onal idempotents e, f ∈ A a ring (e + f)A(e + f) is either ( D D 0 D ) or ( D 0 0 D ) , A is semidistributive, by [8], theorem 14.2.1. Denote Pi = eiiA, i = 1, . . . , 4. Let ϕ : Pi → Pj be a nonzero homomorphism. Then ϕ(eiia) = ϕ(eii)a = ejja0eiia, where a0, a ∈ A and ejja0eii is a nonzero element from ejjAeii = D. Thus d0 = ejja0eii defines a monomorphism. Therefore a ring A is a piecewise domain. But since the right ideal I = (0 D D D) is not projective, A is not right hereditary. Analogously A is not left hereditary. The quiver Q(A) has the following form: • • ??~~~~~~~ • __@@@@@@@ • ??~~~~~~~ __@@@@@@@ (3.4) which is called a diamond. Example 3.2. Let O be a discrete valuation ring and A = Tn(O), where Tn(O) =        O O . . . O O 0 O . . . O O ... ... . . . ... ... 0 0 . . . O O 0 0 . . . 0 O        Jo u rn al A lg eb ra D is cr et e M at h .N. Gubareni, M. Khibina 69 is the ring of all upper n×n-matrices with elements from O. Then Tn(O) is a two-sided Noetherian SPSD-ring and a piecewise domain, but Tn(O) is not hereditary for n ≥ 2. 4. Serial rings with right Noetherian diagonal If A is a serial ring with right Noetherian diagonal, then by theorem 3.3 this diagonal is two-sided Noetherian ring. Therefore it is possible to say in this case simply about Noetherian diagonal. For this ring we can construct the prime quiver which description gives the following theorem. Theorem 4.1. ([12], theorem 2.1). Let A be a serial ring with Noethe- rian diagonal. Then the prime quiver PQ(A) is a disconnected union of cycles and chains. From this theorem and proposition 2.3 we immediately obtain the following statement. Corollary 4.1. Let A be a serial piecewise domain with right Noethe- rian diagonal. Then the prime quiver PQ(A) is a disconnected union of chains. The description of right semihereditary serial rings with right Noethe- rian diagonal was obtained by V.V.Kirichenko, who proved the following theorem. Theorem 4.2. ([16], theorem 2.8). A right semihereditary serial inde- composable ring A with right Noetherian diagonal is up to isomorphism Morita equivalent to a ring H((∆1, n1), . . . , (∆k, nk)) =      A1 A12 . . . A1k O A2 . . . A2k ... ... . . . ... O O . . . Ak      . (4.1) where Aij = Mni×nj (D), ∆i = D or ∆i = Oi a discrete valuation ring with division ring of fractions D, i, j = 1, 2, . . . , k. Moreover Ai = Tni (D) if ∆i = D and Ai = Hni (Oi) if ∆i = Oi. Conversely, all rings of this form are right semihereditary and serial. From theorem 3.4 and corollary 2.3 we immediately obtain the state- ment which is analogous to proposition 3.1. Jo u rn al A lg eb ra D is cr et e M at h .70 Serial piecewise domains Proposition 4.1. If A is a serial piecewise domain with right Noethe- rian diagonal, then its diagonal A is Morita equivalent to the finite direct product of rings of the form D, where D is a division ring, or Hn(O), where O is a discrete valuation ring. Lemma 4.1. Let A be a serial piecewise domain with Noetherian diag- onal which unity is decomposed into two local idempotents. Then A is isomorphic to one of the following rings: (a) T2(D) = ( D D 0 D ) ; (b) H2(O) = H((O, 1), (D, 1)) = ( O D 0 D ) ; (c) H((D, 1), (O, 1), ) = ( D D 0 O ) ; (d) H((O1, 1), (O2, 1)) = ( O1 D 0 O2 ) , where O, O1, O2 are discrete valuation rings with common division ring of fractions D. All these rings are serial piecewise domains and semi- hereditary rings. Proof. We can assume that A is indecomposable and reduced. If A is prime, then A is serial Noetherian prime ring, and, by theorem 3.4, A isomorphic to the ring H2(O). If A is not prime, then, by theorem 2.1, A is isomorphic to a ring ( A1 A12 0 A2 ) , where A1, A2 are prime rings. So, by proposition 4.1, Ai is a division ring or a discrete valuation ring Oi for i = 1, 2. If A1 and A2 are division rings, then the Jacobson radical of A is equal to the prime radical of A, and so it is nilpotent. In this case A is an Artinian ring. And then, by proposition 3.2, A ≃ T2(D). Suppose A1 = O and A2 = D. By theorem 2.1, A12 is an O-D-bimodule, which is not zero, and as a right D module it is isomorphic to D, and as a left O-module is isomorphic to O. Therefore, since A is a serial ring, A12 is a uniserial left O-module which is isomorphic to O. Then analogously as in the proof of lemma 2.7 from [16], we can show that in this case A ≃ H2(O). Analogously we can consider the other cases. Theorem 4.3. For a serial ring A with right Noetherian diagonal the following conditions are equivalent: (i) A is semihereditary; (ii) A is a piecewise domain. Proof. (i) ⇒ (ii) follows from proposition 2.1. Jo u rn al A lg eb ra D is cr et e M at h .N. Gubareni, M. Khibina 71 (ii) ⇒ (i) We can assume that A is indecomposable reduced ring. Since A is a piecewise domain, by theorem 2.1, there exists a decom- position of 1 ∈ A into a sum of mutually orthogonal idempotents 1 = f1 + . . . + ft such that fiAfj = 0 for i > j, all rings Aii = Ai = fiAfi are prime and fiAfi+1 6= 0 for i = 1, . . . , t− 1. Consequently, by proposition 4.1, Ai is either a division ring or a ring of the form Hm(O). Then, by lemma 4.1, Aij ⊆ Mni×nj (D). And so, by theorem 2.1, we have that Aij = Mni×nj (D). Therefore A has the form (4.1), and thus, by theorem 4.2, A is a semihereditary ring. References [1] F.W.Anderson and K.R.Fuller, Rings and Categories of Modules. Second edition, Graduate Texts in Mathematics, Vol.13, Springer-Verlag, Berlin-Heidelberg-New York, 1992. [2] S.U.Chase, A generalization of the ring of triangular matrix, Nagoya Math. J., V.18, 1961, pp.13-25. [3] H.Cartan and S.Eilenberg, Homological algebra. Princeton University Press, Princeton, New York, 1956. [4] D.Eisenbud, P.Griffith, The structure of serial rings, Pacific J. 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[20] T.Nakayama, On Frobeniusean algebras I,II, Ann. of Math., V.40, 1939, pp.611- 633; V.42, 1941, pp.1-21. [21] T.Nakayama, Note on uniserial and generalized uniserial rings, Proc. Imp. Acad. Tokyo, V.16, 1940, pp.285-289. [22] L.A.Skornyakov, When are all modules semi-chained?, Mat. Zametki, V.5, 1969, pp.173-182. [23] L.W.Small, Hereditary rings, Proc. Nat. Acad. Sci. U.S.A., V.55, 1966, pp.25-27. [24] M.L.Teply, Right hereditary, right perfects rings are semiprimary, In: Advances in Ring Theory, 1997, pp.313-316. [25] R.B.Warfield, Serial rings and finitely presented modules, Journal of Algebra, V.37, 1975, pp.187-222. Contact information N. Gubareni Institute of Econometrics & Computer Sci- ence, Technical University of Czȩstochowa, 42-200 Czȩstochowa, Poland E-Mail: nadiya.gubareni@yahoo.com M. Khibina Institute of Engineering Thermophysics, NAS, Kiev, Ukraine E-Mail: marina_khibina@yahoo.com Received by the editors: 25.02.2008 and in final form 25.02.2008.

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