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When Quasi-projective Implies Projective

出版	University of Connecticut, 1981
URL	http://books.google.com.hk/books?id=BJFyNwAACAAJ&hl=&source=gbs_api

註釋Recent investigations in [4], [11], [12], [7] have dealt with the question of the structure of quasi-projective modules. Moreover, the question of module structure for modules with quasi-projective covers, (see [1]), has awaited investigation until quasi-proj ective modules could be sufficiently classified. We have studied both of these questions for modules over a certain class of rings. A Noetherian semi-prime ring A possesses a semi-simple Artinian quotient ring A by Goldie's Theorem. The A-module M is A-torsion free iff the canonical map J:M A M is a monomorphism. Call M A-torsion iff J is the zero map. A ring R is called a pomo-nilring iff (1) N(R) is T-nilpotent (see [2D and (2) R/W(R) is finitely generated by its Noetherian center C. Examples of pomo-nilrings are orders in finite dimensional algebras over a field, and endomorphism rings of torsion free finite rank abelian groups. The first results deal with the structure of superfluous submodules of projective R-modules where R is a pomo-nilring with W(R) = (0). In fact, we prove that if R is a pomo-nilring with N(R) = (0), then superfluous submodules of projective R-modules are finitely generated by R. This allows one to prove that projective covers of R-torsion free modules are isomorphisms. Next, we examine the structure of quasi-projective modules over assorted rings by forcing the existence of projective covers. In the process, it is shown for an arbitrary ring R with center C, that a finitely presented R-module Q is a quasi-projective R-module iff Qj is a quasi-proj ective R^-module for each prime ideal I of C. This and a result of Fuller and Hill [12] proves that finitely presented quasi-projectives over commutative rings are projective modulo their annihilaters. The study of A-torsion free quasi-proj ective modules over Noetherian semi-prime rings A begins with a few results about annihilator ideals of A. It is shown that A-torsion free I-quasi-projective A-modules are projective modulo their annihilaotrs. More-over, if A is prime, the I-quasi-projectives are either projective A-modules or A-torsion modules. The last result we shall mention is: If R is a ring and R/W(R) is a Noetherian prime ring and Q is a finitely generated quasi-proj ective R-module such that Q/W(R)Q is not R/W(R)-torsion, then Q possesses a projective cover over R. Countable pomo-nilrings R present a very pleasant structure. As an example, if Q is a quasi-proj ective R-module and Q/W(R)Q is R/W(R)-torsion free, then Q/W(R)Q is finitely generated R/N(R)- module iff A fl Q/W(R)Q is a finitely generated A-module. [R/N(R) is a Noetherian semi-prime ring with quotient ring A], Moreover, every R/W(R) torsion free quasi-proj ective module is a projective R/W(R)-module iff R/N(R) contains all central idempotents of A. The results of the previous paragraph are true over R/W(R) with "Z-quasi-projective" replaced by "quasi-projective". An application of the above results to quasi-projective covers over a pomo-nilring R produces the following: The Z-quasi-projective covers of R/W(R)-torsion free modules are isomorphisms. If R is countable, "quasi-projective" replaces "Z-quasi-proj ective". Define a pomo-nilring R to be an S-pomo-nil-algebra iff S is a Noetherian integral domain^ R is an S-algebra and R is an S-torsion free module of finite S-rank. Then one notable result is that a finitely generated S-torsion free R-module, M, has a projective cover over R iff M/N(R)M is a projective R/N(R)- module. Moreover, a finitely generated S-torsion R-module Q is quasi-proj ective over R iff is a quasi-proj ective R^ module for each prime ideal I of S. Finally, these results are applied to torsion free abelian groups of finite rank as modules over their endomorphism rings. Let G be such a group and E = End (G). The main theorem is as follows: The following are equivalent: (1) G has projective cover over E. (2) G has a quasi-proj ective cover over E. (3) has a cyclic projective cover over E^ for each prime q 6 Z and G is finitely generated by E. (4) G/W(R)G is a genus summand of E/W(E). (5) G is near isomorphic to MON where M is an E-ring and Hom^ (M, N) (M) = N. (6) G .= H/N where H is a projective End^H)-module and N is a pure subgroup of WCEnd^CH))!!. Furthermore, G is finitely generated by E iff G is quasi-isomorphic to a group H possessing a projective cover over End (H).