Density and equivalence

Quasi-injective and quasi-projective modules over hereditary noetherian prime rings ((hnp)-rings) were studied in [17]. In the present paper we give some applications of the results established in [17]. Kulikov, Kertesz, Prufer, Szele had made basic contributions to the problem of decomposability of abelian p-groups (Fuchs [4]). Kaplansky [9] studied analogous problems for modules over (commutative) Dedekind domains. Let R be an (hnp)-r'mg, which is not right primitive. Using the structure of an indecomposable infective torsion i^-module, established in [17, Theorem 4], some of the basic concepts and results on the decomposability of a torsion abelian group are generalized in Section 2, to modules over R. In particular, it is shown that any non-zero, torsion i^-module has a uniform direct summand, which can be chosen to be of finite length if M is not injective (Theorem 10). As a consequence we get that every divisible i^-module is injective (Corollary 4). This corollary is a special case of Levy [11, Theorem (3.4)]. Some of the analogous results for torsion modules over bounded Dedekind prime rings were established by Marubayashi [12; 13]. His techniques are quite different from those used in this paper. In Section 3, quasi-projective i^-modules are investigated. First of all the concept of primary, torsion i^-module and primary components of a torsion i^-module are introduced. The structure of quasi-projective torsion i^-modules is determined in Theorems 12 through 15. These theorems generalize the structure theorem of a torsion quasi-projective, abelian groups proved by Fuchs and Rangaswamy [5]. Finally non-torsion quasi-projective i?-modules are studied. It is shown in Theorem 16, that any such, reduced i^-module has a non-zero finitely generated projective, direct summand. A result of independent interest is Theorem 15, which states that any quasi-projective finitely generated rightmodule over a prime, right Goldie ring S is projective, whenever it is not a torsion module. All rings considered here are with 1 ^ 0 and all modules are unital right modules. The notations and terminology are essentially the same as in [17] and will be used without comments.

/pdf/modules-over-hereditary-noetherian-prime-rings-4j1uokado2.pdf

Modules over hereditary Noetherian prime rings

Commutative rings in which every finitely generated ideal is quasi-projective

On endomorphism rings of quasiprojective modules

This paper investigates ideal-theoretic as well as homological extensions of the Prufer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prufer conditions.

Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

On quasi-projective modules via relative projectivity

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On quasi-projective modules via relative projectivity