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Monthly Archives: April 2013
Idempotent elements outside the nilradical
Here is a cute exercise in Introduction to Commutative Algebra by Atiyah & Macdonald (A&M), that I recently solved, and felt like writing it up. The problem reads as follows: Exercise 6 (Chapter 1). A ring is such that every … Continue reading
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Tagged idempotent, jacobson radical, maximal ideals, nilradical, prime ideals
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Inverse image of a prime ideal is a prime ideal
Here is one of those problems that invariably shows up in homework for first course on ring theory: Assume and are commutative rings with 1. Prove that if is a ring homomorphism, and is a prime ideal in , then … Continue reading