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Modules finite over endomorphism ring

Part of the Lecture Notes in Mathematics book series (LNM,volume 246)

Abstract

A right R-module M is said to be finendo provided that as a left module over its endomorphism ring it is finitely generated. Let B = End MR. Then Bn → M → 0 is exact for some finite integer n. This implies (and when M is quasi-injective is implied by) the exactitude of 0 → R/annRM → Mn. This implies that a finendo module is quasi iff injective modulo annihilator (1). Similarly for any finitely generated module over a commutative ring. Any finendo quasi-injective module is injective over its biendomorphism ring (17B). A ring R is right Artinian iff every quasi-injective right module is finendo (17). This supplies the converse of a theorem of K. Fuller [69]. Moreover, a ring R is right self-injective iff every finendo faithful (injective) right R-module generates mod-R (18B). Azumaya defined a right PF ring as one such that every faithful right R-module generates mod-R. We characterize a right PF ring as a right self-injective ring such that every faithful (quasi) injective right R-module is finendo (38). Then, if every factor ring is PF, the ring must be right Artinian, hence uniserial (40). In general, a factor ring R/A of a right PF ring is right PF iff the left annihilator of A is Rz = zR for some z ε R (39).

We study Σ-injective modules over a regular ring R, and show that M has Σ-injective hull in mod-R iff M is semisimple, injective, and finendo (16). This generalizes a result of J. Levine [71] assuming that M is simple.

We characterize a prime right Goldie ring as the class of rings over which there exists an indecomposable injective finendo faithful right module E, with no fully invariant nontrivial submodule, such that End ER a field (34). In this case, the biendomorphism ring Q of E is the right quotient ring of R, and E is isomorphic to the unique minimal right ideal of Q (35A). Semiprime Goldie rings can be similarly characterized. Employing a recent characterization of quasi-injective abelian groups by Fuchs, we can describe all the finendo ones: thus, M must be a divisible group containing ℚ, or else M is a torsion group of bounded order the p-components of which are direct sums of isomorphic cyclic groups (16).

A ring is a right V-ring if every simple right R-module is injective. We abbreviate quasi-injective by QI. A ring R is right QI (quasi ⇒ injective) provided that every right QI module is injective. Thus, every right QI ring is a V ring. Moreover, every right QI ring is a right Noetherian (Koehler [70]). Cozzen's example [70] of a right V-domain R is actually a right QI ring, as A. Boyle [71] observed. Any right Goldie right V-ring is a finite product of simple rings (31). This reduces the structure theory for a right V, or QI, ring to simple rings. The latter are characterized as those right Noetherian rings for which the endomorphism ring of any indecomposable injective right module is a field (see the paper of the author [72].

Keywords

  • Local Ring
  • Left Ideal
  • Simple Module
  • Noetherian Ring
  • Endomorphism Ring

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The bibliographic references such as Bass [60] refer to a 1960 paper of Bass listed in the bibliography. References in parenthesis e.g. (1), (17A), etc. refer to numbered propositions, corollaries, or lemma in the text.

The research on this article was supported in part by a grant from the National Science Foundation.

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Faith, C. (1972). Modules finite over endomorphism ring. In: Lectures on Rings and Modules. Lecture Notes in Mathematics, vol 246. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0059565

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  • DOI: https://doi.org/10.1007/BFb0059565

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