We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 12 October 2009
A convention. Now that we have given a formal definition of a ring, we can begin the systematic development of our subject. The rings that we shall consider will all be commutative, and they will all have a unit element. It is therefore convenient to use the word ‘ring’ in a more restricted sense than is customary in modern algebra, and for this reason we lay down the following convention: From now on ‘ring’ will always mean a commutative ring with a unit element. The zero element and the unit element of a ring R will be denoted by 0 and 1 respectively, or, if we are concerned with several rings at the same time, by 0R and IR.
Ideals and their calculus. Let R be a ring (commutative and with a unit element), and let a be a non-empty subset of R, then a is called an ideal of R in all cases where the following two conditions are satisfied:
Whenever a1 and a2 belong to a, then a1 ± a2 both belong to a.
If a ∈ a, then ra ∈ a for all r ∈ R.
A trivial example of an ideal is obtained by taking a to be the whole ring.
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.
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.
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.
