Show that M is an (A/I)-module. Find the submodules.

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SUMMARY

Let A be a ring, I an ideal, and M an A-module with the property that IM=0. M possesses a natural structure as an A/I-module, establishing a clear relationship between the two module types. Additionally, the submodules of M, when viewed as either an A-module or an A/I-module, are identical. This conclusion reinforces the equivalence of module structures under the specified conditions.

PREREQUISITES
  • Understanding of ring theory and A-modules
  • Familiarity with ideals in ring theory
  • Knowledge of module homomorphisms
  • Concept of quotient modules, specifically A/I-modules
NEXT STEPS
  • Study the properties of A-modules and their submodules
  • Explore the concept of quotient modules in depth
  • Learn about the relationship between ideals and module structures
  • Investigate examples of modules with IM=0 to solidify understanding
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Mathematicians, algebraists, and graduate students focusing on module theory and ring theory, particularly those interested in the structural properties of modules over rings.

peteryellow
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Let A be a ring, I an ideal and M an A-module such that IM=0.
Show that M has a natural structure as an A/I-module and vice versa.
Furthermore show that in the situation above the submodules of M considered
as an A-module or A/I-module are the same.
 
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