Ring Homomorphisms 26 Ring Homomorphism and Isomorphism Definition A ring homomorphism φ from a ring R to a ring S is a mapping from R to S that preserves the two ring operations; that is, for all a; b in R, φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b): Definition A ring isomorphism is a ring homomorphism that is both one-to-one and onto. 27 Properties of Ring Homomorphisms Theorem Let φ be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S. 1. For any r in R and any positive integer n, φ(nr) = nφ(r) and φ(r n) = (φ(r))n. 2. φ(A) = fφ(a): a 2 Ag is a subring of S. 3. If A is an ideal and φ is onto S, then φ(A) is an ideal. 4. φ−1(B) = fr 2 R : φ(r) 2 Bg is an ideal of R. 28 Properties of Ring Homomorphisms Cont. Theorem Let φ be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S. 5. If R is commutative, then φ(R) is commutative. 6. If R has a unity 1, S 6= f0g, and φ is onto, then φ(1) is the unity of S. 7. φ is an isomorphism if and only if φ is onto and Ker φ = fr 2 R : φ(r) = 0g = f0g. 8. If φ is an isomorphism from R onto S, then φ−1 is an isomorphism from S onto R. 29 Kernels are Ideals Theorem Let φ be a ring homomorphism from a ring R to a ring S. Then Ker φ = fr 2 R : φ(r) = 0g is an ideal of R. 30 First Isomorphism Theorem for Rings Theorem Let φ be a ring homomorphism from R to S. Then the mapping from R=Ker φ to φ(R), given by r + Ker φ 7! φ(r), is an isomorphism. In symbols, R=Ker φ ∼= φ(R). 31 Ideals are Kernels Theorem Every ideal of a ring R is the kernel of a ring homomorphism of R. In particular, an ideal A is the kernel of the mapping r 7! r + A from R to R=A. 32 Homomorphism from Z to a Ring with Unity Theorem Let R be a ring with unity 1. The mapping φ : Z ! R given by n 7! n · 1 is a ring homomorphism. 33 A Ring with Unity Contains Zn or Z Corollary If R is a ring with unity and the characteristic of R is n > 0, then R contains a subring isomorphic to Zn. If the characteristic of R is 0, then R contains a subring isomorphic to Z. 34 Zm is a Homomorphic Image of Z Corollary For any positive integer m, the mapping φ : Z ! Zm given by x 7! x mod m is a ring homomorphism. 35 A Field Contains Zp or Q Corollary If F is a field of characteristic p, then F contains a subfield isomorphic to Zp. If F is a field of characteristic 0, then F contains a subfield isomorphic to the rational numbers. 36 Field of Quotients Theorem Let D be an integral domain. Then there exists a field F (called the field of quotients of D) that contains a subring isomorphic to D. Proof Idea Let S = f(a; b): a; b 2 D and b 6= 0g and let F be the set of equivalence classes of S under the relation (a; b) ∼ (c; d) iff ad = bc. Write the a equivalence class of (a; b) in S as b . Then F is a field under the usual fraction operations, and the x mapping φ : D ! F given by φ(x) = 1 is an injective ring homomorphism. 37 Acknowledgement Statements of results and some examples follow the notation and wording of Gallian's Contemporary Abstract Algebra, 9th edition. .