New PDF release: A First Course in Rings and Ideals

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.

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, supposéthat 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 isomorpIÚ~;'-: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.

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


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