Summary
Summary Summery of Rings (abstract algebra)
- Course
- Institution
- Book
These are notes by professors which were available online , but later were removed. They are very easy to understand and provided with important examples.
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
Solutions Manual to Accompany Contemporary Abstract Algebra NINTH EDITION Joseph Gallian University of Minnesota Duluth Updated A+
-
Complete Solutions Manual to Accompany Contemporary Abstract Algebra NINTH EDITION Joseph Gallian University of Minnesota Duluth
-
Complete Solution Manual Contemporary Abstract Algebra 9th Edition Gallian Questions & Answers with rationales
Seller
Follow
orionnebula
Content preview
Introduction to RingsSubject: ALGEBRA-III
Semester-IV
Lesson: Introduction to Rings
Lesson Developer: Divya Bhambri
College/Department: St. Stephen’s College,
University of Delhi
Institute of Life Long Learning, University of Delhi Page 1
, Introduction to Rings
Contents
1. INTRODUCTION ............................................................................ 3
2. RINGS ......................................................................................... 3
2.1. Definition and Examples of Rings ............................................... 3
Properties of Rings ............................................................................. 7
2.2. Special types of Rings .............................................................. 8
Boolean Ring .................................................................................. 8
Division Ring ................................................................................... 9
The Ring of Quaternions ................................................................. 10
Field ............................................................................................ 11
3. SUBRINGS ................................................................................. 12
4. Integral Domain ......................................................................... 15
5. Nilpotent and Idempotent elements .............................................. 18
6. Characteristic of a Ring ................................................................ 20
Exercises ........................................................................................... 22
References ........................................................................................ 23
Suggested Readings ........................................................................... 23
Institute of Life Long Learning, University of Delhi Page 2
, Introduction to Rings
1. INTRODUCTION
A ring is an algebraic structure with two binary operations in which arithmetic
operations of addition and multiplication on the set of integers are generalized so
that results from arithmetic are extended to non-numerical objects
like polynomials, matrices and functions.
In 1892, David Hilbert introduced the term "Zahlring" which means number
ring and it was formally published in 1897. The word "Ring" could mean
"association". According to Harvey Cohn, the term ring was used by D. Hilbert that
had the property of "circling its elements directly back to elements of itself”. In
particular, for ring of algebraic integers, all high powers of an algebraic integer can
be written as an integral combination of a known set of lower powers, and thus the
powers "cycle back". For example, if then
and so on; in general, we observe
that is an integral linear combination of and .
2. RINGS
The modern axiomatic definition of (commutative) rings was given by Emmy
Noether in 1920, which led to the development of commutative ring theory.
2.1. Definition and Examples of Rings
Definition: Let be a non-empty set with two binary operations addition and
multiplication (denoted by + and .). If the following axioms are satisfied for all
:
1.
2.
3. –
4.
5.
6.
Then we say that is a ring or we can simply write it as is a ring, in case
there is no scope of confusion about the operations + and . .
If along with these six conditions, elements of also satisfy:
7.
Then we say that is a commutative ring.
Institute of Life Long Learning, University of Delhi Page 3
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller orionnebula. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
81633 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now