1.11. RINGS AND FIELDS77Example 1.11.5.(a)Zis a subring ofQ.(b)The ring of 3-by-3 matrices withrationalentries is a subring ofthe ring of 3-by-3 matrices withrealentries.(c)The set ofupper triangular3-by-3 matrices with real entries is asubring of the ring ofall3-by-3 matrices with real entries.(d)The set ofcontinuousfunctions fromRtoRis a subring of thering ofallfunctions fromRtoR.(e)The set ofrational–valuedfunctions on a setXis a subring ofthe ring ofreal–valuedfunctions onX.A second way in which two rings may be related to one another isby ahomomorphism. A mapfWR!Sbetween two rings is said tobe ahomomorphismif it “respects” the ring structures. More explicitly,fmust take sums to sums and products to products. In notation,f .aCb/Df .a/Cf .b/andf .ab/Df .a/f .b/for alla; b2R. A bijectivehomomorphism is called anisomorphism.Example 1.11.6.(a)The mapfWZ!Zndefined byf .a/DOEaŁis a homomor-phism of rings, becausef .aCb/DOEaCbŁDOEaŁC
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