🔥This ℤ×ℤ Question Shocked UPSC IFoS 2025 😱 | Prime vs Maximal Ideal Explained

Описание: 🔥This ℤ×ℤ Question Shocked UPSC IFoS 2025 😱 | Prime vs Maximal Ideal Explained

🚨 UPSC IFoS 2025 Mathematics Optional – Modern Algebra PYQ 🚨
Q) Show that in the ring ZxZ, (i) the ideal S = {(a, 0): a ∈ Z} is a prime ideal but
not maximal and (ii) the ideal T = {(m, n) ∈ ZxZ:3|n} is a maximal ideal.
Asked By UPSC IFoS 2025 Mains Examination Maths Optional Paper -2 Modern Algebra For 10+5 Marks

This question from UPSC IFoS 2025 shocked many serious aspirants! 😱
In the ring ℤ × ℤ, one ideal turns out to be prime but NOT maximal, while another becomes maximal — a classic UPSC trap 🔥
This video explains one of the most confusing and concept-testing questions asked in UPSC Indian Forest Service (IFoS) 2025 from Ring Theory — the famous ℤ × ℤ ideal problem 😱
Many serious aspirants lost marks because they could not clearly differentiate between a Prime Ideal and a Maximal Ideal.
This video removes that confusion once and for all 🔥

👉 In this video, Soni Sir explains:
✅ Why S={(a,0)} is prime but not maximal
✅ Why T={(m,n):3∣n} is a maximal ideal
✅ How quotient rings decide maximality
✅ How to write exam-scoring proofs in UPSC answers

🎯 What You Will Learn in This Video
✅ How to identify ideals in product rings
✅ Conceptual difference between Prime vs Maximal Ideals
✅ Role of quotient rings in deciding maximality
✅ Why ℤ is not a field, but ℤ₃ is a field
✅ How UPSC sets traps in Modern Algebra questions
✅ Exam-ready proof writing technique for full marks

📌 Must-watch for UPSC CSE & IFoS Maths Optional aspirants
📌 Concepts repeatedly tested in Modern Algebra

🎯 Why This Video is Important?
✔ Frequently repeated PYQ concept
✔ Differentiates Prime vs Maximal (very confusing area)
✔ Helps avoid silly mistakes in exam

🎓 About the Instructor
👨‍🏫 Soni Sir
Founder & Mentor – Ganit Vishay Gurukul | Higher Educational Mathematics Institute
📍 Offline Classes: Nagpur
💻 Online Classes: PAN India
📘 Specialized in UPSC Maths Optional, IFoS, CSIR-NET, MSc Entrance

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✔ PYQ-focused teaching
✔ Concept-first approach
✔ Board-friendly & exam-oriented proofs
✔ Proven results in higher mathematics

🔥 This single question can upgrade your entire Ring Theory preparation.
📚 Study Smart. Think Deep. Crack UPSC.
— Team Ganit Vishay Gurukul 💪

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🧑‍🏫 Taught by :- Soni Sir | Expert in Advanced Mathematics

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