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Keywords:
chain rings; serial rings; uniserial modules
chain rings; serial rings; uniserial modules
Summary:
The purpose of this paper is to provide a criterion of an occurrence of uncountably generated uniserial modules over chain rings. As we show it suffices to investigate two extreme cases, nearly simple chain rings, i.e. chain rings containing only three two-sided ideals, and chain rings with ``many'' two-sided ideals. We prove that there exists an $\omega_{1}$-generated uniserial module over every non-artinian nearly simple chain ring and over chain rings containing an uncountable strictly increasing (resp. decreasing) chain of right (resp. two-sided) ideals. As a consequence we describe right steady serial rings.
References:
[BBT] Bessenrodt Ch., Brungs H.H., Törner G.: Right Chain Rings. Part 1. Schriftenreihe des Fachbereich Mathematik Universität Duisburg (1990).
[C] Cohn P.M.: Free Rings and Their Relations. Academic Press Conder, New York (1971). MR 0371938 | Zbl 0232.16003
[D1] Dubrovin N.I.: An example of a chain prime ring with nilpotent elements. Math. USSR, Sb. 48 (1984), 437-444. MR 0691988 | Zbl 0543.16003
[D2] Dubrovin N.I.: Chain domains. Moscow Univ. Math. Bull. 35 2 (1980), 56-60. MR 0570590 | Zbl 0456.16001
[EGT] Eklof P.C., Goodearl K.R., Trlifaj J.: Dually slender modules and steady rings. Forum Math. (1997), 9 61-74. MR 1426454 | Zbl 0866.16003
[F] Facchini A.: Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules. Birkhäuser Basel (1998). MR 1634015 | Zbl 0930.16001
[FS] Fuchs L., Salce L.: Modules over Valuation Domains. Marcel Dekker New York and Basel (1985). MR 0786121 | Zbl 0578.13004
[P1] Puninski G.: Some model theory over nearly simple uniserial domain and decomposition of serial modules. J. Pure Appl. Algebra (2001), 21 319-337. MR 1852123
[P2] Puninski G.: Some model theory over an exceptional uniserial rings and decomposition of serial modules. J. London Math. Soc., Ser. II. (2000), 64 2 311-326. MR 1853453
[P3] Puninski G.E., Tuganbaev A.A.: Rings and Modules (Kol'ca i moduli). Moskva, 1998 (in Russian). MR 1641739
[P4] Puninski G.: Serial Rings. Kluwer Academic Publ. Dordrecht (2001). MR 1855271 | Zbl 1032.16001
[T] Trlifaj J.: Almost $\star$-modules need not be finitely generated. Comm. Algebra (1993), 21 2453-2462. MR 1218507
[Z] Žemlička J.: Steadiness is tested by a single module. Contemporary Mathematics (2001), 273 301-308. MR 1817172 | Zbl 0988.16003
[ZT] Žemlička J., Trlifaj J.: Steady ideals and rings. Rend. Sem. Mat. Univ. Padova (1997), 98 161-172. MR 1492975
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