[1] J.L. Alperin, Local Representation Theory, Cambridge University Press, 1986.
[2] F.W. Anderson, KR. Fuller, Rings and Categories of Modules. 2-nd edition, Springer, 1992.
[3] M. Auslander, I. Reiten, O. Smale, Representation Theory of Artinian Algebras, Cambridge University Press, 1996.
[4] K Bessenrodt, H.H. Brungs, G. Tomer, Right chain rings. Part 1, Schriftenreihe des Fachbereich Mathematik, Universitat Duisburg, 1990.
[5] K Bessenrodt, H.H. Brungs, G. Tomer, Right chain rings. Part 2a, Schriftenreihe des Fachbereich Mathematik, Universitat Duisburg, 1992.
[6] K Bessenrodt, H.H. Brungs, G. Tomer, Right chain rings. Part 2b, Schriftenreihe des Fachbereich Mathematik, Universitat Duisburg, 1992.
[7] G. Birkhoff, Lattice Theory. 3-rd edition, Amer. Math. Soc., Providen-ce, R. I., 1967.
[8] N. Bourbaki, Alqebre Commutative, Hermann, 1961-1965. Chapitre 1-7.
[9] R. Camps, W. Dicks, On semilocal rings, Israel J. Math ., 81 (1993), 203-221.
[10] R. Camps, A. Facchini , Priifer rings that are endomorphism rings of artinian modules, Commun. Algebra, 22 (1994), 3133-3157.
[11] R. Camps, A. Facchini, G. Puninski, Serial rings that are endomor-phism rings of artinian modules, Rings and Radicals, Pitman, Res. Not. in Math., 346 (1996), 141-159.
[12] R. Camps, P. Menal, Power cancellation for artinian modules, Com-mun. Algebra, 19 (1991), 2081-2095.
[13] A.W . Chatters, Serial rings with Krull dimension, Glasgow Math . J., 32 (1990), 71-78.
[14] A.W . Chatters, C.R. Hajarnavis, Rings with Chain Conditions, Pit-man, 1980.
[15] C.C . Chang, H.J. Keisler, Model Theory, Studies in Logic and the Foundations of Mathematics, 73 (1973) .
[16J P.M. Cohn, On the free product of associative rings, Math . Z., 71 (1959), 380-398.
[17J P.M. Cohn, Free Rings and Their Relations, Academic Press, 1971. [18] P. Crawley,B.Jonsson, Refinements for infinite direct decompositions
of algebraic systems, Pacif. J. Math ., 14 (1964), 797-855.
[19J Ch. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, John Wiley &Sons, 1962.
[20] F. Dischinger, B. Muller, Einreihig zerlegbare artinsche Ringe sind selbstdual, Arch . Math., 43 (1984), 132-136.
[21] Yu. Drozd, On generalized uniserial rings, Mathemat. Zametki, 18 (1975), 707-710.
[22] Yu. Drozd, V.V. Kirichenko , Finite Dimensional Algebras, Springer, 1994.
[23J N. Dubrovin, The rational closure of group rings of left-ordered groups, Schriftenreihe des Fachbereich Mathematik, Universitat Duisburg, 1994.
[24J Nguen Viet Dung, A. Facchini , Weak Krull-Scmidt for infinite direct sums of uniserial modules, J. Algebra,193 (1997), 102-121.
[25] Nguen Viet Dung, A. Facchini, Direct summands of serial modules, J. Pure Appl. Algebra, 113 (1998), 93-106.
[26] D. Eisenbud, P. Griffith, Serial rings , J. Algebra,17 (1971), 389-400. [27] D. Eisenbud, P. Griffith, The structure of serial rings ,Pacij. J.Math.,
36 (1971), 109-121.
[28] P. Eklof, I. Herzog, Model theory of modules over a serial ring, Ann. Pure Appl. Math., 72 (1995), 145-176.
[29] P. Eklof, G. Sabbagh, Model completions and modules, Ann. Math . Logic, 2 (1971), 251-295.
[30] K. Faith, Algebra: Rings, Modules and Categories I, Springer, 1973. [31] K. Faith, Algebra II. Ring Theory ,Springer, 1976.
[32] A. Facchini, Decomposition of algebraically compact modules, Pacific J. Math., 116 (1987), 25-37.
[33] A. Facchini, Relative injectivity and pure injective modules over Priifer rings, J. Algebra, 110 (1987), 380-406.
[34] A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc., 348 (1996), 4561-4576 .
[35] A. Facchini , Module Theory : Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Birkhauser, 1998. [36] A. Facchini, D. Hebrera, L. Levy, P. Vamos, Krull-Schmidt fails for
artinian modules, Proc. Amer. Math. Soc., 123 (1995), 3587-3592. [37] A. Facchini, G. Puninski, ~-pureinjective modules over serial rings,
Abelian groups and modules, Kluwer Acad. Publishers, 1995, 145-162. [38] A. Facchini, G. Puninski, Classical localizations in serial rings,
Com-mun. Algebra, 24 (1996), 3537-3559.
[39] A. Facchini, L. Salce, Uniserial modules: sums and isomorphisms of subquotients, Commun. Algebra, 18 (1990), 499-517.
[40] J.W. Fisher, Nil sub rings of endomorphism rings of modules, Proc. Amer. Math. Soc., 34 (1972), 75-78.
[41J 1. Fuchs, L.Salce, Modules over Valuation Domains, Lect . Notes Pure Appl. Math., 97 (1985).
[42J K Fuller, Generalized uniserial rings and their Kuppisch series , Math. Z., 106 (1968), 248-260.
[43J KR. Fuller, Artinian Rings (a supplement to Rings and Categories of Modules), Notas de Matematica 2, University Murcia, 1989.
[44J S. Garavaglia, Direct product decomposition of theories of modules, J. Symb. Logic, 44 (1979), 77-86.
[45J S. Garavaglia, Decomposition of totally transcendental modules, J. Symb. Logic., 45 (1980), 155-164.
[46J A.I. Generalov, Proper classes of short exact sequences over right noetherian serial rings, Predel'nye Theoremy Teorii Veroyatnostej, 1 (1986), 87-102.
[47J A.W. Goldie , Torsion-free modules and rings, J. Algebra, 1 (1964), 268-287.
[48J KR. Goodearl, Von Neumann Regular Rings, Pitman, 1979.
[49J R. Gordon, J.e. Robson, Krull Dimension , Mem . Amer. Math. Soc., 133 (1973).
[50J O.E. Gregul', V.V. Kirichenko, On semihereditary serial rings . Ukrai-nskij Matem. Zhurnal, 39 (1987) , 156-161.
[51J G. Gratzer, Gen eral Lattice Theory, Academie-Verlag, 1978.
[52J F. Guerriero,B.Muller, Prime ideals in serial rings, Commun. Algebra, 27 (1999), 4531-4544.
[53J J.K Haak, Self-duality and serial rings, J. Algebra, 59 (1979), 345-363.
[54J J.K Haak, Serial rings and sub direct products, J.Pure Appl. Algebra, 30 (1983), 157-165.
[55J D. Hebrera, A. Shamsuddin, Modules with semilocal endomorphism ring, Proc. Amer. Math . Soc., 123 (1995) , 3593-3600.
[56] I. Herzog, The Auslander-Reiten translate, Abelian Groups and Non-comm. Rings, Collect . Pap. Mem. RB. Warfield, 1991, 153-165. [57] I. Herzog, Elementary duality for modules, Trans. Amer. Math . Soc.,
340 (1993), 37-69.
[58] I. Herzog, A test for finite representation type, J. Pure Appl. Algebra, 95 (1994), 151-182.
[59] G.J. Janusz, Indecomposable modules for finite groups, Ann. Math., 89 (1969), 209-241.
[60] C.D. Jensen, H. Lenzing, Model Theoretic Algebra, Gordon and Breach, 1989.
[61] I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.
[62] I. Kaplansky, Infinite Abelian Groups, Ann. Arbor, Michigan, 1954. [63] F . Kasch, Moduln und Ringe, Stuttgart, 1977.
[64] V.V. Kirichenko, Generalized uniserial rings, Preprint, Kiev, 1975. [65] V.V. Kirichenko, Right noetherian rings over which finitely generated
modules are serial, Doklady Akad. Nauk Ukrain. SSR. Ser . A, 1(1976), 9-12.
[66] V.V. Kirichenko, Generalized uniserial rings, Matem. Sbornik, 99
(1976), 559-581.
[67] V.V. Kirichenko, On serial hereditary and semihereditary rings , Za-piski Leningradskogo Otdeleniya MIAN, 114 (1982), 137-147.
[68] G. Krause, T .H. Lenagan, Transfinite powers of the Jacobson radical, Commun. Algebra,7 (1979), 1-8.
[69] H. Kuppisch, Beitrage zur Theorie nichthalbeinfacher Ringen mit Min-imalbedingung, J. Reine Angew. Math., 201 (1959), 100-112.
[70] H. Kuppisch, Uber eine Klasse von Ringen mit Minimalbedingung, Arch. Math ., 17 (1966), 20-35.
[71] H. Kuppisch, Einreihige Algebren tiber einem perfekten Korper, J.
[72] T.H. Lenagan, Reduced rank in rings with Krull dimension, Lect. Notes Pure Appl. Math., 51 (1979), 123-131.
[73] J.C. McConnell, J .C. Robson, Noncommutative Noetherian R ings, John Wiley & Sons, 1988.
[74] G. Michler, Structure of semiperfect hereditary noetherian rings, J. Algebra, 13 (1969) , 327-344.
[75] A.P. Mishina, L.A. Skornyakov, Abelian Groups and Modules, Moscow, Nauka, 1969.
[76] S.H. Mohamed, B.J. Muller, Continuous and Discrete Modules, Cam-bridge University Press, 1990.
[77] I. Monk, Elementary-recursive decision procedures, Doctoral thesis, Berkeley, 1975.
[78] I.Murase, On the structure of generalized uniserial rings. I,Sci. Papers College Gen. Educ ., Univ. Tokyo, 13 (1963), 1-22.
[79] I. Murase, On the structure of generalized uniserial rings. II, Sci. Pa-pers College Gen. Educ .,Univ. Tokyo, 13 (1963), 131-158.
[80] 1. Murase, On the structure of generalized uniserial rings. III, Sci. Papers College Gen. Educ., Univ. Tokyo, 14 (1964) , 11-25.
[81] 1. Murase, Generalized uniserial group rings. I, Sci. Papers College Gen. Educ., Univ . Tokyo, 15 (1965), 15-28.
[82] 1. Murase 1. Generalized uniserial group rings. II, Sci. Papers College Gen. Educ., Univ . Tokyo, 15 (1965), 111-128.
[83] B.J. Muller, The structure of serial rings , Methods in Module Theory, Marcel Dekker, 1992, 249-270.
[84] B.J. Muller, Goldie-prime serial rings, Contemp. Math., 130 (1992), 301-310.
[85] B.J. Muller, Ore sets in serial rings, Commun. Algebra, 25 (1997) , 3137-3146.
[86] B.J. Muller, S. Singh, Uniform modules over serial rings. II,Lect. Not es in Math., 1448 (1989), 25-32.
[87] B.J. Muller, S. Singh, Uniform modules over serial rings, J. Algebra, 144 (1991), 94-109.
[88] T . Nakayama, Note on uniserial and generalized uniserial rings, Proc. Imp. Acad. Tokyo, 16 (1940), 285-289.
[89] T. Nakayama, On Frobenius algebras. II, Ann. Math.,42 (1941), 1-21. [90] R.S. Pierce, Associative Algebras, Graduate Texts in Mathematics,
Springer, 1982.
[91] M. Prest, Model Theory and Modules , Cambridge University Press, 1988.
[92] G.E. Puninski, Superdecomposable pure injective modules over com-mutative valuation rings, Algebra i Logika, 31 (1992), 655-671. [93] G.E. Puninski, Indecomposable pure injective modules over uniserial
rings, Trudy Moskovskogo Matem. Obshchestva, 56 (1995), 176-191. [94] G.E. Puninski, Pure injective modules over right noetherian serial
rings, Commun. Algebra,23 (1995), 1579-1592.
[95] G.E. Puninski, Serial Krull-Schmidt rings and pure injective modules, Fundamental'naya i Prikladnaya Matem. , 1 (1995) , 471-490.
[96] G.E. Puninski, Finitely presented modules over uniserial rings , Fun-damental 'naya i Prikladnaya Matem.,3 (1997), 631-633.
[97] G. Puninski, Cantor-Bendixson rank of the Ziegler spectrum over a commutative valuation domain, J. Symb. Logic,64 (1999), 1512-1518. [98] G. Puninski, Some model theory over a nearly simple uniserial do-main and decompositions of serial modules, J. Pure Appl. Algebra, to appear.
[99] G. Puninski, Some model theory over an exceptional uniserial ring and decompositions of serial modules, J. Lond. Math. Soc., to appear. [100] G. Puninski, M. Prest, Ph. Rothmaler, Rings described by various
purities, Commun. Algebra, 27 (1999), 2127-2162.
[101] G. Puninski, R. Wisbauer, M. Yousif, On p-injective rings , Glasgow Math . J.,37 (1995), 373-378.
[102] V. Puninskaya, Vaught's conjecture for modules over a serial ring, J. Symb. Logic,65 (1999), 155-163.
[103] G. Reynders, Ziegler spectra of serial rings with Krull dimension, Com-mun. Algebra, 27 (1999) , 2583-2611.
[104] J.e.Robson, Idealizers and hereditary noetherian prime rings, J. Al-gebra, 22 (1972), 45-81.
[105] Ph. Rothmaler, A trivial remark on purity, Proc. 9-th Easter Conf. in Model Theory, Seminarber. 112 (1991), Humboldt-Univ. , 127.
[106] Ph. Rothmaler, Mittag-Leffler modules and positive atomicity, Pre-print, 1994.
[107] Ph. Rothmaler, Mittag-Leffler modules, J. Pure Appl. Algebra, 88 (1997), 227-239.
[108] L.H. Rowen, Ring Theory, Academic Press, 1991.
[109] L. Salce, Valuation domains with superdecomposable pure injective modules, Lect. Notes Pure Appl. Math., 146 (1993), 241-245.
[110] S. Singh, Serial right noetherian rings , Canad. J. Math ., 36 (1984) , 22-37.
[111] L.A. Skornyakov, When are all modules serial?, Matem. Zametki, 5 (1969), 173-182.
[112] B. Stenstrom, Rings of Quotients, Springer, 1975.
[113] W. Stephenson, Modules whose lattice of submodules is distributive, Proc. Lond. Math. Soc., Ser. 3., 28 (1974) , N2, 291-310.
[114] A.A. Tuganbaev, Rings with flat right ideals and distributive rings, Matem. Zametki, 38 (1985) , 218-228.
[115] A.A. Tuganbaev, Distributive rings and endodistributive modules, Uk-rainskij Matem . Zhumal, 38 (1986), 63-67.
[116] A.A. Tuganbaev, Hereditary rings, Matem. Zametki, 41 (1987) , 303-312.
[117] A.A. Tuganbaev, Sem idistributive Rings and Modules, Mathematics and its applications, Kluwer, 449 (1998).
[118] M.H. Upham, Serial rings with rightKrull dimension one, J. Algebra, 109 (1987), 319-333.
[119] RB. Warfield, Purity and algebraic compactness for modules, Pac ij. J. Math. , 28 (1969), 699-719.
[120] R.B. Warfield, Decomposability of finitely presented modules, Proc. Amer. Math. Soc., 25 (1970), 167-172.
[121] RB. Warfield, Exchange rings and decomposition of modules, Math. Ann., 199 (1972), 31-36.
[122] RB. Warfield, Serial rings and finitely presented modules, J. Algebra, 37 (1975), 187-222.
[123] RB. Warfield, Bezout rings and serial rings, Commun. Algebra, 7 (1979), 533-545.
[124] J . Waschbiisch, Self-duality of serial rings, Commun. Algebra, 14 (1986), 581-589.
[125] R Wisbauer, Foundations of Module and Ring Theory , Gordon and Breach, 1991.
[126] M.H. Wright, Serial rings with right Krulldimension one. II, J. Alge-bra,117 (1988), 99-116.
[127] M.H. Wright, Krull dimension in serial rings, J. Algebra, 124 (1989), 317-328.
[128] M.H. Wright, Links between right ideals of a serial rings with Krull dimension, Lect. Notes in Math., 1448 (1989), 33-40.
[129] M.H. Wright, Certain uniform modules over serial rings are uniserial, Commun. Algebra, 17 (1989), 441-469.
[130] M.H. Wright, Uniform modules over serial rings withKrulldimension, Commun. Algebra, 18 (1990), 2541-2557.
[131] M.H. Wright, Prime serial rings with Krull dimension, Commun. Al-gebra, 18 (1990), 2559-2572.
[132] M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic, 26
[133] W. Zimmermann, Rein injektive direkte Summen von Moduln, Com-mun. Algebra, 5 (1977), 1083-1117.
[134] B. Zimmermann-Huisgen, W. Zimmermann, Algebraically compact rings and modules, Math. Z., 161 (1978), 81-93.
[135] B.V . Zabavskij, N.Ya. Komarnizkij, Distributive domains with ele-mentary divisors, Ukrainskij Matem. Zhurnal, 42 (1990), 1000-1004.
ann2(I)(R) second annihilator of a left ideal I in a ring 188 R
ann2(R)(I) second annihilator of a right ideal I in a ring 188 R
ann(m)(R) annihilator of an element m of a module in a 4 ring R
ann(M)(r) annihilator of an element r of a ring in a mod- 4 ule
C(I) the set r E R, such that r
+
I is a nonzero 41 divisor in R/IE(M) injective envelope ofM 3
G(m,n) a special semigroup 98
G(R) Goldie radical of a serial ring R 36
Hn,m(V) special right Noetherian serial ring over a uni- 89
serial domainV
Hn(V) special Noetherian serial ring over a uni-serial 89
domain V
i"-+j linkage notion in the quiver of a serial ring 80 I* complement of a right ideal I in a serial ring 139
(I, J) pair of (left and right) ideals in a serial ring 136
Inv(CP,1/J) invariant sentence 132
J* complement of a left ideal J in a serial ring 139
J*jI special pp-type in a serial ring 139
J(a)(R) transfinite power of the Jacobson radical of a 5 ring R
Jac(R) Jacobson radical ofR 1
Kdim(M) the Krull dimension ofM 5
L(R) lattice of pp-formulas in one variable over R 126
M=N elementary equivalence of M and N 132
M
f=
ep(m) a pp-formulaip is satisfied by an element m 132 of a module MNil(R) nil radical of a ring R 2
N(p) pure injective envelope of a pp-typep 131
P
lattice of pp-types over a ring 128P~Q linkage notion between prime ideals in a serial 36
ring
1l"(R) the intersection of prime ideals containing an 52 idempotent of a serial ring R
P localizable system of prime or completely 43,42 prime ideals in R
Pm(R) prime radical ofR 2
Qp p-adic integers 120
R;. diagonal ring of a semi-perfect ringR 8
R;.j R;.-Rj bimoduleeiRej for a semi-perfect ring 8
R
Sing(M) singular submodule of M 4
T(M) torsion submodule of a module M over a uni- 19 serial domain
Tn(D) ring of upper triangular matrices over a skew 19 field D
U(R) group of invertible elements ofR 1
Z(p) localization of integers via prime idealpZ 19
annihilator second 188 chain
in the linkage graph of a serial ring 39
in the quiver of a serial ring 80
circle
in the linkage graph of a serial ring 39
in the quiver of a serial ring 80
clique
of prime ideals in a serial ring 39 cone on a group 18 criterion
*
(of consistency of a pp-type) 139**
(of consistency of a pp-type) 140 Ziegler 132 domain 2 nearly simple 174 dimension Goldie 4 Krull 5 duality Prest 129 element 224 torsion 179 elements linked by pp-formula 130 envelope injective 3 projective 3 pure injective 131 e-pairof ideals in a serial ring 39 formula
basic 151
positive primitive (pp-) 123 dual 129
e- 136
left positive primitive (left pp-) 129 RD- (relative divisibility) 133 free realization of a pp-formula 127 group left ordered 18 ideal completely prime 2 Goldie 4 prime 2 semi-prime2 idempotent abelian 163 implication trivial 124
lemma common denominator 125 Drozd 10 Prest 124 localizable system of completely prime ideals 42 of prime ideals 43 module Artinian 1 balanced 73 Bezout 164 coherent 174 distributive 133 endo-Artinian 198 endo-distributive 134 endo-uniserial134 finitely presented 9 fp-injective 133 injective 3 Krull-Schmidt 28 local 6 locally coherent 174 Noetherian 1 non-principal~j 67 non-singular 4 p-injective 91 pp-atomic 173 pp- uni-serial 147 principal~j 67 projective 3 pure injective 129 pure projective 130 quasi-small 186 serial 9 singular 4 small 3 super-decomposable 151, 147 torsion 179 torsion-free 179 uniform 3 uni-serial 9 ~-pure injective 198 modules elementarily equivalent 132 morphism irreducible 192 predecessor
of a prime ideal in a serial ring 36 of a simple module 91 quiver 80 radical Goldie 36 Jacobson 1 nil 2 prime 2 ring abelian regular 163 Artinian 1
with a.c.c. on right annihilators 4 basic 8 coherent 174 d- 100 distributive 133 duo 32 exceptional 187
finite representation type 18 Goldie 4 group 17 hereditary 3 Krull-Schmidt 32 link-free 70 local 6 nearly simple 187 Noetherian 1 non-singular 4
prime 2 QF- (quasi-Frobenius) 18 quasi-matrix 105 of quotients classical 16 with respect to a localizable system 42 (Von Neumann) regular 163
of typeI 163 semi-duo 11 semigroup 17 semi-hereditary 3 semi-local 6 semi-perfect 7 semi-primary 2 semi-prime 2 semi-primitive 1 semisimple Artinian 2 serial 9 T- 64 tiled 73 uni-serial 9
upper triangular matrices over a division ring 19 over a uni-serial ring 89 set maximal Ore 17 right denominators 16 s-94 weak Ore 17 series Kuppisch 95 sentence invariant 132 sequence admissible 96 split epimorphism 2 monomorphism 2 subgroup pp-definable (pp-) 123 type-definable 128 submodule elementary 132 essential 3 torsion 179 successor
of a prime ideal in a serial ring 36 of a simple module 91 theorem Crawley-Jenssen-Warfield 172 Drozd-Warfield 21 Goldie 4 Kaplansky 172 Kirichenko 89 Kirichenko-Warfield 89 Krull-Schmidt 6 type e- 136
finitely generated pp-type173 indecomposable pp-type 132 J*
/1
139pp- 127
super-decomposable pp-type 156
