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In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that〔Artin, p. 353〕〔Atiyah and Macdonald, p. 2〕〔Bourbaki, p. 102〕〔Eisenbud, p. 12〕〔Jacobson, p. 103〕〔Lang, p. 88〕 * ''f''(''a'' + ''b'') = ''f''(''a'') + ''f''(''b'') for all ''a'' and ''b'' in ''R'' * ''f''(''ab'') = ''f''(''a'') ''f''(''b'') for all ''a'' and ''b'' in ''R'' * ''f''(1''R'') = 1''S''. (Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. On the other hand, neglecting to include the condition ''f''(1''R'') = 1''S'' would cause several of the properties below to fail.) If ''R'' and ''S'' are rngs (also known as ''pseudo-rings'', or ''non-unital rings''), then the natural notion〔Hazewinkel et al. (2004), p. 3. Warning: They use the word ''ring'' to mean rng.〕 is that of a rng homomorphism, defined as above except without the third condition ''f''(1''R'') = 1''S''. It is possible to have a rng homomorphism between (unital) rings that is not a ring homomorphism. The composition of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms (cf. the category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism. == Properties == Let be a ring homomorphism. Then, directly from these definitions, one can deduce: * ''f''(0''R'') = 0''S''. * ''f''(−''a'') = −''f''(''a'') for all ''a'' in ''R''. * For any unit element ''a'' in ''R'', ''f''(''a'') is a unit element such that . In particular, ''f'' induces a group homomorphism from the (multiplicative) group of units of ''R'' to the (multiplicative) group of units of ''S'' (or of im(''f'')). * The image of ''f'', denoted im(''f''), is a subring of ''S''. * The kernel of ''f'', defined as , is an ideal in ''R''. Every ideal in a commutative ring ''R'' arises from some ring homomorphism in this way. * The homomorphism ''f'' is injective if and only if . * If ''f'' is bijective, then its inverse ''f''−1 is also a ring homomorphism. In this case, ''f'' is called an isomorphism, and the rings ''R'' and ''S'' are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished. * If there exists a ring homomorphism then the characteristic of ''S'' divides the characteristic of ''R''. This can sometimes be used to show that between certain rings ''R'' and ''S'', no ring homomorphisms can exist. * If ''Rp'' is the smallest subring contained in ''R'' and ''Sp'' is the smallest subring contained in ''S'', then every ring homomorphism induces a ring homomorphism . * If ''R'' is a field and ''S'' is not the zero ring, then ''f'' is injective. * If both ''R'' and ''S'' are fields, then im(''f'') is a subfield of ''S'', so ''S'' can be viewed as a field extension of ''R''. * If ''R'' and ''S'' are commutative and ''P'' is a prime ideal of ''S'' then ''f''−1(''P'') is a prime ideal of ''R''. * If ''R'' and ''S'' are commutative and ''S'' is an integral domain, then ker(''f'') is a prime ideal of ''R''. * If ''R'' and ''S'' are commutative, ''S'' is a field, and ''f'' is surjective, then ker(''f'') is a maximal ideal of ''R''. * If ''f'' is surjective, ''P'' is prime (maximal) ideal in ''R'' and , then ''f''(''P'') is prime (maximal) ideal in ''S''. Moreover, *The composition of ring homomorphisms is a ring homomorphism. *The identity map is a ring homomorphism (but not the zero map). *Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings. * For every ring ''R'', there is a unique ring homomorphism . This says that the ring of integers is an initial object in the category of rings. * For every ring ''R'', there is a unique ring homomorphism , where 0 denotes the zero ring (the ring whose only element is zero). This says that the zero ring is a terminal object in the category of rings. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ring homomorphism」の詳細全文を読む スポンサード リンク
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