By David M. Burton

ISBN-10: 0201007312

ISBN-13: 9780201007312

This quantity is designed to function an creation to the fundamental rules and strategies of ring idea. it really is meant to be an expository textbook, instead of a treatise at the topic. The mathematical heritage required for a formal realizing of the contents isn't huge. We think that the common reader has had a few past touch with summary algebra yet continues to be really green during this admire. as a result, approximately every little thing herein might be learn by means of somebody accustomed to uncomplicated group-theoretic techniques and having a nodding acquaintance with linear algebra. the extent of fabric may still turn out compatible for complex undergraduates and starting graduate scholars.

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Example text

The reader will experience no difficulty in'showing that the principal ideal generated bya is given by ,', = {na + ra + as + ¿ More generally, let {J;} be an arbitrary indexed collection of ideals of 1, and is the ideal of R whose members are aH possible finite sumS of elements from the various ideals I i : \L '¿ 1, = {¿ afia, ~:jJ. R. , The sum of this collection may be deno~ed by . ,' finite + a2 + ... + a"la,EIJ, I i ; aH but a , + Proof If 1 = {O}, the theorem is triviaHy true, since' the zero ideal {O} is the principal ideal generated by O.

Llthough it might seem to be quite special, will serve us in good stead when we develop the theory of Noetherian rings. Theorem 2-5. 1), 2) 1 :, L J¡ = n (I :, J¡), 3) 1 :r(JK)'= (1 :rK) :rJ. 24 IDEALS AND THEIR OPERA TIONS FIRST COURSE IN RINGS AND IDEALS Proo! Concerning (1), we have (n 1J:rJ = = {a E RlaJ ~ n 1¡} = {a E RlaJ ~ 1¡ for all i} n{aERlaJ ~ 1¡} c= n(l¡:r J )· . With an eye to proving(2), notice that the inc1usion J¡ ~ ~ J¡ implies a~~ J;) ~ 1 if and only if aJ¡ ~ 1 for all i; thus, 25 often referred to as the pseudo-in verse of a.

Suppose that f is a homomorphism from the ring R onto the ring R'. We have already observed that each ideal l of the ring R determines an ideal f(I) of the ring R'. It goes without saying that ring theory would be considerably simplified ifthe ideals of R were in a one-to-one correspondence with those of R' in this manner. Unfortunately, this need not be the case. The difficulty is refiected in the fact that if l and J are two ideals of R with l ~ J ~ l + kerJ, thenf(I) = f(J). The quickest way to see this is to notice tbat f(l) ~ f(J) ~ f(l + kerf) = f(I) + f(kerf) 31 IDEALS AND THEIR OPERA TIONS FIRST COURSE IN RINGS AND IDEALS = f(l), from which we conclude that all the inclusions are actually equalities.

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

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