Hopfian and co-Hopfian modules over Artinian rings

Abstract

If R is a ring with 1, we call a unital left R-module M Hopfian (co-Hopfian) in the category of left R-modules if any epic (monic) R-module endomorphism of M is an automorphism. In the case R is a commutative Noetherian ring, we use a result of Matlis to characterize those injective R-modules that are co-Hopfian, and to characterize those that are Hopfian when R is also reduced. We show that if R is a commutative Artinian principal ideal ring, then an R module M is Hopfian (co-Hopfian) if and only if M is finitely generated if and only if its injective envelope E(M ) is Hopfian (co-Hopfian) if and only if E(M ) is finitely generated. We note that the "finite uniserial type" problem poses an obstacle to establishing this result for an arbitrary Artinian principal ideal ring.

co-Hopfian Modules

“…This theorem generalizes the co-Hopfian part of an observation made in [14,Rem 3.6] for the case that R is Artinian. If R is not Artinian, then the module E in the theorem may not be Hopfian, since E(R/m) may not be Hopfian (see the discussion in [14, §2]).…”

“…It would be nice to know what conditions allow a converse, that is, when M co-Hopfian implies E(M ) is. We saw in [14] that the converse holds if R is an Artinian P IR. We will say more in this regard in Section 5.…”

“…Recall that an R-module is finitely cogenerated if its injective envelope is a finite sum of injective envelopes of simple modules [14,Def 3.4]. Using the two theorems of Matlis, we can rephrase Theorem 1.3 as follows.…”

“…One might naturally wonder for which 0-dimensional R is it the case that the only co-Hopfian R-modules are the finitely generated ones. We discussed this question in [14] and showed that if R is an Artinian P IR, then an R-module M is co-Hopfian if and only if it if finitely generated.…”

“…Note that all finitely generated modules over a commutative ring are bi-Hopfian if and only if the ring is 0dimensional. In [14], we showed that over a commutative Artinian principal ideal ring, a module is Hopfian (co-Hopfian) if and only if it is finitely generated (among other equivalent conditions).…”