6.2. HOMOMORPHISMS AND IDEALS271In particular, a ring homomorphism is a homomorphism for the abeliangroup structure ofRandS, so we know, for example, that'.x/D'.x/and'.0/D0.Even ifRandSboth have an identity element1, it is not automatic that'.1/D1. If we want to specify that this is so,we will call the homomorphism aunitalhomomorphism.Example 6.2.2.The map'WZ!Zndefined by'.a/DOEaŁDaCnZis a unital ring homomorphism. In fact, it follows from the definition ofthe operations inZnthat'.aCb/DOEaCbŁDOEaŁCOEbŁD'.a/C'.b/,and, similarly,'.ab/DOEabŁDOEaŁOEbŁD'.a/'.b/for integersaandb.Example 6.2.3.LetRbe any ring with multiplicative identity1. The mapk7!k 1is a ring homomorphism fromZtoR. The map is just the usualgrouphomomorphism fromZto the additive subgrouph1igenerated by1; see Example2.4.7. It is necessary to check thath1iis closed under mul-tiplication and that this map respects multiplication; that is,.m 1/.n 1/Dmn 1. This follows from two observations:
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Abelian group, Identity element, ring homomorphism