An R-module M is called weakly co-Hopfian if any injective endomorphism of M is essential. The class of weakly co-Hopfian modules lies properly between the class of co-Hopfian and the class of Dedekind finite modules. Several equivalent conditions are given for a module to be weakly co-Hopfian. Being co-Hopfian, weakly co-Hopfian, or Dedekind finite are all equivalent conditions on quasi-injective modules. Some other properties of weakly co-Hopfian modules are also obtained. The ring R is said to be right strong stably finite if all the finitely generated free right R-modules are weakly co-Hopfian. We shall characterize such rings and show that they are stably finite and satisfy the right strong rank condition. Examples show that stably finite rings and rings with the right strong rank conditions need not be strong stably finite. Both weakly co-Hopfian and right strong stably finite are Morita invariants, although the right and left strong stably finite are different properties. The class of commutative rings and the class of rings with finite right uniform dimension are proper subclasses of the class of right strong stably finite rings. We shall also investigate conditions that are relevant to weakly co-Hopfian modules. Equivalent statements are found on a ring to have all its finitely generated right modules weakly co-Hopfian. ᮊ 2001 Academic Press WEAKLY CO-HOPFIAN MODULESRings will have unit elements and modules will be unitary. The terminolw x ogy not defined here may be found in 6 . Let R be a ring and M a right
For a semi-projective retractable module MR with endomorphism ring S, we prove u.dim MR= u.dim SS, and find necessary and sufficient conditions on M in order that S be respectively semiprime, right nonsingular, finitely cogenerated, cocyclic, or weakly co-Hopfian. Precise descriptions of the right singular ideal of S and the socle of M are given, and in addition if S is a semiprime ring, it is shown that MR is FI-extending if and only if SS is FI-extending.
An endoprime R-module M is a nonzero module whose nonzero fully invariant submodules are faithful over End (MR). Such modules have prime endomorphism rings. The class of endoprime modules properly contains the class of prime modules in the sense of Zelmanowitz [5], and the class of prime modules with condition ( * ) of Wisbauer [4]. While over commutative rings, endoprime modules are prime, in general, prime and endoprime are different conditions. A nonzero right R-module M is a direct sum of isomorphic simple modules if and only if each nonzero element of σ[M ] is an endoprime module. Various properties of endoprime modules including their Morita invariance, are obtained, and the structure of endomorphism rings of certain endoprime modules are determined. Generalities on endoprime modulesThroughout all rings have nonzero identity elements and modules are unital. There are several equivalent statements defining a prime ring, but when generalizations of these statements to modules are considered one does not necessarily end up with equivalent statements. Recall that the ring R is prime if and only if for all a, b in R, aRb = 0 implies a = 0 or b = 0. Thus, one may call a nonzero right R-module M prime if mRb = 0, m ∈ M , b ∈ R implies m = 0 or M b = 0, or equivalently all nonzero submodules of M have the same annihilator. We know that R is a prime ring if and only if any nonzero ideal has zero left annihilator, or stated otherwise, any nonzero fully invariant submodule of R R is faithful over the endomorphism ring End (R R )R. This latter property is generalized to modules as follows:Definition. A nonzero right R-module M is called endoprime if any nonzero fully invariant submodule of M R is faithful as a left module overImmediate examples of endoprime modules are critical modules and uniform nonsingular modules, since nonzero endomorphisms of such modules Key words and phrases: endoprime, weakly co-Hopfian, quotient like.
By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module R M is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) virtually semisimple if R R is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring R is left completely virtually semisimple if and only ifwhere k, n 1 , ..., n k ∈ N and each D i is a principal left ideal domain. Moreover, the integers k, n 1 , ..., n k and the principal left ideal domains D 1 , ..., D k are uniquely determined (up to isomorphism) by R. * The research of the first author was in part supported by a grant from IPM (No. 94130413). This research is partially carried out in the IPM-Isfahan Branch.†
Let R be a ring with identity and let M be a unitary right R-module. Then M is essentially compressible provided M embeds in every essential submodule of M. It is proved that every non-singular essentially compressible module M is isomorphic to a submodule of a free module, and the converse holds in case R is semiprime right Goldie. In case R is a right FBN ring, M is essentially compressible if and only if M is subisomorphic to a direct sum of critical compressible modules. The ring R is right essentially compressible if and only if there exist a positive integer n and prime ideals P i (1 i n) such that P 1 ∩ · · · ∩ P n = 0 and the prime ring R/P i is right essentially compressible for each 1 i n. It follows that a ring R is semiprime right Goldie if and only if R is a right essentially compressible ring with at least one uniform right ideal.
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