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De nition 1.2. A subset Sof a ring Ris called a subring of Rif Sitself is a ring with respect to the laws of composition of R Proposition 1.3. A non-empty subset Sof a ring Ris a subring of Rif and only if a b2Sand ab2Swhenever a;b2S Prof.o If Sis a subring then obviously the given condition is satis ed. Conversely, suppose that the
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4. Local rings 103 5. The Prime Ideal Principle of Lam and Reyes 103 6. Minimal Primes 108 7. An application to unit groups 110 Chapter 5. Examples of Rings 111. Pushing and pulling via a localization map 149 3. The fibers of a morphism 151 4. Commutativity of localization and passage to a quotient 152 5. Localization at a prime ideal 152 6.
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The kernel of the homomorphism ϕ is nZ. Example 16.21. Let C[a, b] be the ring of continuous real-valued functions on an interval [a, b] as in Example 16.5. For a fixed α ∈ [a, b], we can define a ring homomorphism ϕα: C[a, b] → R by ϕα(f) = f(α). This is a ring homomorphism since.
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If the 4-ring 'were proper we would have four primary options (A, B) (A, C) (C, D) (B, D) for both sides, which can be satisfied only by putting A. D and B. C on opposite sides. But this is contrary to the con-tingent options, hence proper 4-rings are excluded from minimal maps. The exclusion of proper 1-rings, 2-rings, and 3-rings is simpler.
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Types of morphisms defined by properties of ring maps. In this section we study what properties of ring maps allow one to define local properties of morphisms of schemes. Definition 29.14.1. Let be a property of ring maps. We say that is local if the following hold: For any ring map , and any we have . For any rings , , any , , and any ring map.
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