Download A First Course in Rings and Ideals by David M. Burton PDF

By David M. Burton

This quantity is designed to function an creation to the elemental rules and methods of ring concept. it truly is meant to be an expository textbook, instead of a treatise at the topic. The mathematical heritage required for a formal knowing of the contents isn't really huge. We think that the typical reader has had a few earlier touch with summary algebra yet remains to be really green during this recognize. as a result, approximately every little thing herein may be learn by way of an individual conversant in uncomplicated group-theoretic recommendations and having a nodding acquaintance with linear algebra. the extent of fabric may still end up compatible for complicated undergraduates and starting graduate scholars.

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Sample 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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