Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-06T07:48:04.460Z Has data issue: false hasContentIssue false
This chapter is part of a book that is no longer available to purchase from Cambridge Core

CHAPTER I - Definitions and Fundamental Properties

Neal H. McCoy
Affiliation:
Smith College
Get access

Summary

Definition of a ring. Let us consider a set R of elements a, b, c, …, such that for arbitrary elements a and b of R there is a uniquely defined sum a + b and product ab (sometimes written as a·b) which are also elements of R. The words addition and multiplication, as in the ordinary usage of elementary algebra, will be respectively associated with the operations of forming a sum, or a product, of elements of R. Such a set is said to be a ring if addition and multiplication have the five properties listed below, it being assumed that a, b, and c are arbitrary elements of R, either distinct or identical:

P1 · a + (b + c) = (a + b) + c (associative law of addition);

P2 · a + b = b + a (commutative law of addition);

P3 · The equation a + x = b has a solution x in R;

P4 · a(bc) = (ab)c (associative law of multiplication);

P5 · a(b + c) = ab + ac, (b + c)a = ba + ca (distributive laws).

The importance of the concept of ring follows primarily from the fact that there are so many important mathematical systems of quite different types which are rings according to the above definition. Naturally, what they all have in common are the properties used in the definition of a ring, together with any properties which are logical consequences of these.

Type
Chapter
Information
Rings and Ideals , pp. 1 - 30
Publisher: Mathematical Association of America
Print publication year: 1948

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×