By David M. Burton

ISBN-10: 0201007312

ISBN-13: 9780201007312

This quantity is designed to function an advent to the elemental principles and methods of ring concept. it's 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 large. We think that the typical reader has had a few previous touch with summary algebra yet continues to be rather green during this recognize. consequently, approximately every little thing herein may be learn via somebody conversant in easy group-theoretic ideas and having a nodding acquaintance with linear algebra. the extent of fabric should still end up appropriate for complicated undergraduates and starting graduate scholars.

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**Download e-book for kindle: A First Course in Rings and Ideals by David M. Burton**

This quantity is designed to function an creation to the fundamental rules and strategies of ring conception. it's 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 really huge. We suppose that the common reader has had a few past touch with summary algebra yet continues to be quite green during this appreciate.

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**Additional resources for A First Course in Rings and Ideals**

**Example text**

All ) + (b l , bz, ... ,b,,) (al + b l , al + bz, ... , a" + b,,) and (al' az, ... , all)(bl,b z' ... , b,,) = (a l b l , azb z, ... , a"b"). The ring so obtained is caBed the external direct sum of R l , R z, , .. , R" and is convenientIy written R = RI Rz R". ) In brief, the situation is this: An external direct sum is a new ring constructed from a given set of rings, and an interna! direct sum is a representation of a given ring as a sum of certain of its ideals. The connection between these two types of direct sums will be made c1ear in the next paragraph.

First, iffis a one-to-one function andf(a) ,;;,. 0= feO), then a = O, whence ker f = {O}. On the other hand, supposÃ©that the kernel consists exactly of O. , fea - b) = fea) - f(b) = O,'; which means that a - bE ker f. Since ker f = {O},\vemust have a - b = O, or a = b, making f a one-to-one function. ,:< : Two rings R and R' are said to be isomorpIÃš~;'-:denoted by R ~ R', if there exists an isomorphism fro'm the ring R ontP the ring R ':.. t~on of a furistion from ~ne particular'ring to another, let us remark that iff: R --+ R'isaone~to-one, onto; homomorphic mapping, the function f -1: R" --+ Ralso enjoys these properties.

This disconcerting situation could be remedied by either demanding that kerf = {O} or else narrowing our view to coilsider only ideals l with ker f ~ l. In either event, it follows that l ~ J ~ l + ker f = l and, in consequence, l = J. The first of the restrictions just cited has the effect of making the function f one-to-one, in which case R and R' are isomorphic rings (and it then comes as no surprise to find their ideals in one-to-one correspondence). The second possibility is the subject of our next theorem.

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

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