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RING HOMOMORPHISM

5.2.1 Definition. Let and be commutative rings. A function ∶ → is called a ring homomorphism if Section 5.2 (i) , for all , ∈ ; (ii) , for all , ∈ ; and

Ring Homomorphisms (iii) 11. A ring homomorphism that is one‐to‐one and onto is called a ring isomorphism. If there is a ring isomorphism from onto , we say that is isomorphic to , and write ≅ . A ring isomorphism from the commutative onto itself is called and automorphism of .

INVERSES AND COMPOSITION OF EXAMPLES OF HOMOMORPHISMS RING ISOMORPHISMS • Natural Projection. The mapping ∶ → 5.2.2 Proposition. given by , for all ∈ , is a ring (a) The inverse of a ring isomorphism is a homomorphism that is onto but not one‐to‐ ring isomorphism. one. (b) The composite of two ring isomorhisms • Natural Inclusion. Let be a commutative is a ring isomorphism. ring, and define ∶ → by , for all ∈ . That is, is the constant polynomial . This is a ring homomorphism that is one‐to‐one but not onto.

BASIC PROPERTIES OF RING EVALUATION MAPPING HOMOMORPHISMS Let be a subfield of the field . For any element ∈ , we can define a function 5.2.3 Proposition. Let ∶ → be a ring homomorphism. Then ∶ → by letting , for each ∈ . Then the requirements for a (a) 00; ring homomorphism are satisfied. Since the (b) for all ∈ ; polynomials in are evaluated at , the homomorphism is called an evaluation (c) is a subring of . mapping.

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KERNEL OF A RING HOMOMORPHISM PROPERTIES OF THE KERNEL

5.2.4 Definition. Let ∶ → be a ring 5.2.5 Proposition. Let ∶ → be a ring homomorphism. The set homomorphism. ∈ | 0 (a) If , ∈ ker and ∈ , then , , and belong to ker . is called the kernel of , denoted by ker . (b) The homomorphism is an isomorphism if and only if ker 0and .

FUNDAMENTAL HOMOMORPHISM GENERALIZED SETTING FOR THE THEOREM FOR RINGS EVALUATION MAPPING 5.2.6 Theorem. Let ∶ → be a ring 5.2.7 Proposition. Let and be commutative homomorphism. Then ⁄ ker ≅ . rings, let ∶ → be a ring homomorphism, and let be any element of . Then there exists a unique ring homomorphism ∶→ such that for all ∈ , and .

COMPONENT‐WISE ADDITION AND ROOTS OF POLYNOMIALS MULTIPLICATION

Let be a subring of the ring , and let 5.2.8 Proposition. Let ,,…, be commutative rings. The set of ‐tuples ,,…, such that ∈ for each is a ∶ → be the inclusion mapping. If ∈ S, commutative ring under the following addition and then the ring homomorphism ∶→ multiplication: defined by Proposition 5.2.7 should be thought ,,…, ,,…, , ,…, of as an evaluation mapping, since , ,…, ⋅ , ,…, , ,…, , for any polynomial ∈ . If 0, then we say that is a root of the For ‐tuples ,,…, and ,,…, . polynomial .

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DIRECT SUM CHARACTERISTIC OF A RING

5.2.9 Definition: Let ,,…, be commutative rings. The set of ‐tuples ,,…, such that 5.2.10 Defintion. Let be a commutative ∈ for each , under the operations of ring. The smallest positive integer such that component‐wise addition and multiplication is called ⋅ 1 0 is called the characteristic of , the direct sum of the commutative rings ,,…,, denoted by char . and is denoted by If no such positive integer exists, then is ⊕ ⊕⋯⊕ . said to have characteristic zero.

INTEGRAL DOMAINS AND CHARACTERISTIC 5.2.11 Proposition. An integral domain has characteristic 0 or , for some prime number .

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