Generating Rings with Ideals: The Possibilities and Implications

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SUMMARY

This discussion centers on the properties of rings in relation to ideals, specifically whether all rings must be generated by ideals. It confirms that while every ring has at least two ideals, {0} and R, not all elements within a ring need to be generated by ideals. The conversation highlights that if an ideal contains the element 1, it generates the entire ring, indicating a significant characteristic of the ring. Additionally, the example of Gaussian integers illustrates that some rings can be generated by a limited number of ideals.

PREREQUISITES
  • Understanding of ring theory and its fundamental concepts
  • Familiarity with ideals in algebraic structures
  • Knowledge of Gaussian integers and their properties
  • Basic comprehension of simple rings and their characteristics
NEXT STEPS
  • Explore the properties of simple rings and their implications in algebra
  • Study the role of ideals in generating rings, focusing on examples like Gaussian integers
  • Investigate the relationship between elements and ideals in various ring structures
  • Learn about the significance of the element 1 in ideal generation within rings
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Mathematicians, algebra students, and researchers interested in ring theory and the role of ideals in algebraic structures.

Simfish
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Do all rings have to be generated by ideals? Or can some rings come without ideals?

Can some elements in rings be generated by ideals (in ways that other elements of rings are untouched by ideals?)

If ALL of a ring's elements are generated by ideals, is there something special about the ring? (for example, the ring of Gaussian integers is completely generated by around 6 ideals IIRC).

If an element in a ring is hit on by two different ideals, is there anything wrong with that?
 
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every ring R always has 2 ideals, namely {0} and R

i don't understand your other questions
 
No proper ideals = Simple rings.

If I remember correctly, if an ideal contains 1, then that ideal generates the ring or something like that.

Can't think of anything else at this hour.
 

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