Generating Rings with Ideals: The Possibilities and Implications

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In summary, the conversation discusses the relationship between rings and ideals. It questions whether all rings must be generated by ideals or if some rings can exist without ideals. The concept of elements in rings being generated by ideals is also brought up, and whether this has any significance for the ring itself. The idea of simple rings being those without proper ideals is mentioned, as well as the possibility of an ideal containing the element 1 and thus generating the entire ring. The conversation concludes with the acknowledgement that there may be other factors to consider in this topic.
  • #1
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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  • #2
every ring R always has 2 ideals, namely {0} and R

i don't understand your other questions
 
  • #3
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.
 

1. What is the purpose of generating rings with ideals?

The purpose of generating rings with ideals is to study and understand the structure and properties of rings. Ideals provide a way to generalize the concept of divisibility in rings, which can help in solving equations and proving theorems.

2. How are rings with ideals generated?

Rings with ideals are generated by taking a set of elements and defining operations (addition and multiplication) on them that satisfy certain properties, such as associativity and distributivity. The ideal is then a subset of the ring that is closed under these operations.

3. What are the implications of generating rings with ideals?

The implications of generating rings with ideals are vast and varied. One major implication is the ability to study and classify different types of rings, such as commutative or non-commutative rings. This also allows for the development of abstract algebraic structures that have applications in various fields of mathematics and science.

4. Can rings with ideals be generated from any set of elements?

No, not all sets of elements can generate a ring with ideals. The set must satisfy certain conditions, such as closure under the defined operations, in order for a ring with ideals to be generated.

5. What are some real-world applications of generating rings with ideals?

Generating rings with ideals has applications in many areas of mathematics and science, such as cryptography, coding theory, and algebraic geometry. It also has practical applications in engineering and computer science, such as in signal processing and error correction algorithms.

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