MATH 436 Notes: Factorization in Commutative Rings

نویسنده

  • Jonathan Pakianathan
چکیده

Proposition 1.1. Let f : R1 → R2 be a homomorphism of rings. If J is an ideal of R2, then f (J) is an ideal of R1 containing ker(f) and furthermore f(f(J)) ⊆ J . Now let f : R1 → R2 be an epimorphism of rings. If J is an ideal of R2 then f(f (J)) = J . If I is an ideal of R1 then f(I) is an ideal of R2. Furthermore we have I ⊆ f (f(I)) = I + ker(f) and thus I = f(f(I)) if I contains ker(f). Thus if f : R1 → R2 is an epimorphism of rings, then there is a bijective correspondence:

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تاریخ انتشار 2010