By David M. Burton

ISBN-10: 0201007312

ISBN-13: 9780201007312

This quantity is designed to function an advent to the elemental rules and strategies of ring idea. it really is meant to be an expository textbook, instead of a treatise at the topic. The mathematical historical past required for a formal knowing of the contents isn't broad. We suppose that the typical reader has had a few previous touch with summary algebra yet remains to be particularly green during this appreciate. as a result, approximately every little thing herein may be learn by way of an individual acquainted with simple group-theoretic innovations 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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**Extra resources for A First Course in Rings and Ideals**

**Sample text**

Proof. We already know that the trivjal subring {O} forms an ideal of R'. Since ker f = f-l(O), the conclusion follows from the last corollary. IDEALS AND THEIR OPERA TIONS 29 'The kernel of a hcimomorphism may be viewed as a measure of the extent to which the homomorphism fails to be one-to-one (hence, fails to be ~ isomorphism). In more concrete terms, we have Theorem 2-10. A homomorphism f from a ring R into a ring R' is an isomorphism if and only ifker f = {O} . Proof. First, iffis a one-to-one function andf(a) ,;;,.

Two cornrnents regarding part (3) of the aboye theorern are in order: First, it is evident that f(a)l = (Jg)(a) = f(a)g(a) f(a - b) f(a) = f(a1) = f(a)f(l) = f(a) + f( - b) = f(a) - f(b). In short, any ring hornornorphisrn preserves differences as well as sums and products. The next theorem indicates the algebraic nature of direct and inverse images of subrings under homomorphisms. Among other things, we shall see that iffis a homomorphism from the ring R into the ring R', thenf(R) forms a subring of R'.

Thatfis a hornornorphisrn follows directly frorn the definition of the operations üi Zn: f(a + b) = [a + b] f(ab) = [ab] = = [a] +n [b] [aln[b] = f(a) +nf(b), = f(a)·nf(b). Example 2-6. In the ring rnap(X, R), define La to be the function which assigns to each fE rnap(X, R) its value at a fixed elernent a E X; in other words, La(J) =:= f(a): Then La is a hornornorphisrn frorn rnap(X, R) into R, known as the evaluation homomorphism at a. We need only observe that Proo! Frorn f(O) = f(O + O) == f(O) + f(O), we obtain f(O) = O.

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