Transcription of RING HOMOMORPHISMS AND THE ISOMORPHISM …
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RING HOMOMORPHISMS AND THE ISOMORPHISM …
ring HOMOMORPHISMS AND THE ISOMORPHISM THEOREMSBIANCA VIRAYWhen learning about groups it was helpful to understand how different groups relate toeach other. We would like to do so for rings, so we need some way of moving betweendifferent (R,+R, R)and(S,+S, S)be rings. A set map :R Sis a( ring )homomorphismif(1) (r1+Rr2) = (r1) +S (r2)for allr1,r2 R,(2) (r1 Rr2) = (r1) S (r2)for allr1,r2 R, and(3) (1R) = simplicity, we will often write conditions(1)and(2)as (r1+r2) = (r1) + (r2)and (r1r2) = (r1) (r2)with the particular addition and multiplication : (R,+, ) (S,+, )is a ring homomorphism then : (R,+) (S,+)isa group any ring andS Ris a subring, then the inclusioni:S Ris a that :Q Mn(Q), (a) = .. a is a ring a field and leta F. Prove that :F[x] F, (f(x)) =f(a)is a ring Zbe a positive integer. Prove that :Z Z, (a) =naisnota ring a ring and letIbe an ideal. Prove that :R R/I, (r) =r+Iis a ring if the following maps are (1) : M2(R) R, ((a bc d))=a(2) : M2(R) R, (A) = Tr(A)(3) : M2(R) R, (A) = det(A)The three definining properties of a ring homomorphism imply other important :R Sbe a ring homomorphism.
1. Kernel, image, and the isomorphism theorems A ring homomorphism ’: R!Syields two important sets. De nition 3. Let ˚: R!Sbe a ring homomorphism. The kernel of ˚is ker˚:= fr2R: ˚(r) = 0gˆR and the image of ˚is im˚:= fs2S: s= ˚(r) for some r2RgˆS: Exercise 9. Let Rand Sbe rings and let ˚: R!Sbe a homomorphism. Prove that ˚is
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