Finiteness conditions for projective and injective modules

Does Hopkins' theorem extend to projective modules, i.e., are projective Artinian modules Noetherian? An example is given to answer this question in the negative; however, we show that the answer is affirmative for certain large classes of projective modules. Dually, are injective Noetherian modules Artinian? Again the answer is negative; nevertheless, we provide an affirmative answer for certain classes of injective modules. Introduction. It is well known that the endomorphism ring of a module which is both Artinian and Noetherian is semiprimary. The author has noted [7] that the endomorphism rings of both projective Artinian and injective Noetherian modules are semiprimary. This raised the following questions: Do there exist projective Artinian modules which are not Noetherian, or could it be true that Hopkins' theorem [11, p. 132] extends to projective modules? Dually, do there exist injective Noetherian modules which are not Artinian [8], [9, p. 378]? In ?1 we prove that the endomorphism rings of both projective Artinian and injective Noetherian modules are semiprimary. We give an example of a projective Artinian non-Noetherian module in ?2 and show that Hopkins' theorem does extend to a projective Artinian R-module M in each one of the following cases: (a) R is commutative, (b) R is hereditary, or (c) M is a generator in the category of R-modules. An example of an injective Noetherian non-Artinian module is given in [14]. In ?3 we prove that if M is an injective Noetherian R-module where R is a ring with polynomial identity which satisfies the ascending chain condition on annihilators of submodules of M, then M contains an essential submodule which is Artinian. From this it follows that injective Noetherian modules over commutative rings are Artinian. Presented to the Society, March 3, 1971 under the title Endomorphism rings of modules; received by the editors September 29, 1972 and, in revised form, December 27, 1972. AMS (MOS) subject classifications (1970). Primary 16A50, 16A52, 16A46; Secondary 16A64, 16A38, 18E10.