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Prime and Maximal Ideals Video Lecture | CSIR NET Crash Course for Mathematics - CSIR NET Mathematics
FAQs on Prime and Maximal Ideals Video Lecture - CSIR NET Crash Course for Mathematics - CSIR NET Mathematics
| 1. What is the difference between prime and maximal ideals? |  |
Ans. Prime ideals are those which are not proper subset of any other ideal except the whole ring itself, whereas maximal ideals are those which are maximal among all proper ideals in a ring.
| 2. How are prime ideals related to integral domains? | |
Ans. In an integral domain, every prime ideal is a maximal ideal. This property is not true for general rings.
| 3. Can a ring have multiple maximal ideals? | |
Ans. Yes, a ring can have multiple maximal ideals. For example, the ring of integers has infinitely many maximal ideals.
| 4. Are all maximal ideals prime ideals as well? | |
Ans. No, not all maximal ideals are prime ideals. However, in a principal ideal domain (PID), every maximal ideal is a prime ideal.
| 5. How are prime and maximal ideals useful in ring theory and algebraic geometry? | |
Ans. Prime and maximal ideals play a crucial role in ring theory by providing tools to study properties of rings, such as factorization. In algebraic geometry, they help in understanding the geometry of algebraic varieties through the concept of the Nullstellensatz.
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Mar 07, 2025
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