Download Alexandre Grothendieck's EGA V by Blass P., Blass J. PDF

By Blass P., Blass J.

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Example text

If 1 S;; cent R, then there is a unique homomorphic extension of f to all of R. Prooj As a start, we choose the element u E 1 so ihat f(u) = 1. Since 1 constitutes an ideal of R, the product au wi11lie in the set 1 for each choice of a E R. It is therefore possible to define a new function g: R -¡. R' by setting g(a) f(au) for all a in R. If the element a happens to belong to 1, then f(au), = f(a)f(u) = f(a)1 fea), showing that g actual1y extends the original functionj The next thing to confirm lS that both ring operations are _preserved by g.

L. We now list sorne of the structural features preserved under hornornorphisrns. Theorem 2-7. Letfbe a hornomorphism frorn the ring R into the ring R'. Then the following hold: 1) f(O) = O, Proo! To obtain the first part of the theorem, recall that, by definition, the imagef(S) = {f(a)la E S}. Now, suppose thatf(a) andf(b) are arbitrary elements of f{S). Then both a and b belong to th~ set S, as do a -:- b and ab (S being a subring of R). Hence, f(a) - f(b) = f(a - b) Ef(S) and 2) f( - a) = - f(a) for all a E R.

If the element a happens to belong to 1, then f(au), = f(a)f(u) = f(a)1 fea), showing that g actual1y extends the original functionj The next thing to confirm lS that both ring operations are _preserved by g. The case of addition is fairly obvious: if a, b E R, then g(a + b) = f(a = f(au) As a preliminary step to demonstrating that g also preserves multiplication, notice that f(ab)u Z ) = f(abu)f(u) f(abu). From 1rus we are able to conclude that it is evident that R X {O} constitutes a subring of R x Z.

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