• DOI:10.1090/gsm/188
• Corpus ID: 51946380

Introduction to Algebraic Geometry

@article{Cutkosky2018IntroductionTA,
  title={Introduction to Algebraic Geometry},
  author={Steven Dale Cutkosky},
  journal={Graduate Studies in Mathematics},
  year={2018}
}

• S. Cutkosky
• Published 31 May 2018
• Mathematics
• Graduate Studies in Mathematics

Recall that a map Φ : U → V of sets is a 1-1 correspondence (a bijection) if and only if Φ has an inverse map; that is, a map Ψ : V → U such that Ψ ◦Φ = idU and Φ ◦Ψ = idV . Lemma 1.3. Let π : R→ S be a surjective ring homomorphism, with kernel K. 1. Suppose that I is an ideal in S. Then π−1(I) is an ideal in R containing K. 2. Suppose that J is an ideal in R such that J contains K. Then π(J) is an ideal in S. 3. The map I 7→ π−1(I) is a 1-1 correspondence between the set of ideals in R and the set of ideals in S which contain K. The inverse map is J 7→ π(J). 4. The correspondence is order preserving: for ideals I1, I2 in S, I1 ⊂ I2 if and only if π(I1) ⊂ π(I2). 5. For an ideal I in S, I is a prime ideal if and only if π−1(I) is a prime ideal in R. 6. For an ideal I in S, I is a maximal ideal if and only if π−1(I) is a maximal ideal in R.