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Fully Prime Modules and Fully Semiprime Modules

Profile image of John A BeachyJohn A Beachy

2019

https://doi.org/10.4134/BKMS.B190864
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17 pages

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Abstract

Fully prime rings (in which every proper ideal is prime) have been studied by Blair and Tsutsui, and fully semiprime rings (in which every proper ideal is semiprime) have been studied by Courter. For a given module M , we introduce the notions of a fully prime module and a fully semiprime module, and extend certain results of Blair, Tsutsui, and Courter to the category subgenerated by M . We also consider the relationship between the conditions (1) M is a fully prime (semiprime) module, and (2) the endomorphism ring of M is a fully prime (semiprime) ring.

Key takeaways
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  1. Fully prime modules and fully semiprime modules extend concepts from ring theory to module theory.
  2. The text establishes a relationship between fully prime modules and their endomorphism rings.
  3. A finitely generated left Artinian ring has fully prime modules that are semisimple and homogeneous.
  4. Theorem 5.8 confirms that End R (P) is fully prime for finitely generated projective modules over fully prime rings.
  5. Courter's sixteen equivalent conditions for fully semiprime rings find analogs in the category σ[M].

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