Review of Rings

We recall the main notions of ring theory: definitions of rings, ideals, quotients, R/I field iff I maximal and R/I domain iff I prime. Elements which are prime, irreducible, zero divisors, nilpotent. Euclidean rings, PID, UFD and relations with examples and counter examples. The set of nilpotent elements is an ideal and is the intersection of all prime ideals. The radical of an ideal. The Jacobson radical. An element x is in the Jacobson radical iff 1-xy is a unit for every y. Ring homomorphisms and ideals. Extensions and contractions. Definition of normal rings. UFD implies normal. Definition of local ring. The rings $\mathbb{Z}[\sqrt{-d}]$ for $d\geq 0$ , which are UFD and which not. Short discussion about the rings $\mathbb{Z}[\sqrt{d}]$ for $d\geq 0$ and $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$ . We show that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID non an euclidean ring.

References: Chap 1 of Atiyah-Macdonald, PDF

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