# MA231: Abstract Algebra I

Unit 3: Fundamentals of Rings   Rings are “larger” structures than groups, because they are sets with two binary operators instead of one and because they have more properties than groups.  Technically, a ring is a commutative group, under addition, with the added property of being associative under multiplication.  The distributive property connects the two operations.  The study of rings began as a study of generalizations on addition and multiplication of integers.  Interestingly, the desire to generalize structures of number sets and operations came from the interest in solving Fermat’s Last Theorem.

We begin the unit with a more formal definition of rings and look at ring properties.  We will then look at commutative and non-commutative rings (note that if a ring is commutative under multiplication, it is defined as a “commutative ring”).  The study of these two classes of rings arose independently and mathematicians tended not to work on both at the same time.  Though most research has focused on commutative rings, non-commutative rings have become an important area of study over the last sixty years.  We do wish to familiarize ourselves with non-commutative rings, but we will not spend too much time studying them, for such a study could realistically command an entire course of its own.

Next, we will move into the concept of ideals, which is based on the work of J.  W.  R.  Dedekind.  It was Dedekind and his ideals that first moved abstract algebra from number theory to structure theory.  Dedekind was looking for the properties of “ideal complex numbers,” but he generalized his findings to abstract structures rather than working strictly with number sets.  He defined an ideal as a subring whose elements, when multiplied by *any** element of the larger ring, result in an element of the subring.  That is, for a ring R and a subring S, if the product rs is in S for every r in R and s in S, then S is an ideal.  A maximal ideal is the “largest proper ideal” of a ring.  That is, if R is a ring and I is an ideal of R, then I is maximal, if there are no* largerproper ideals in *R.  Another way of approaching maximal ideals is if **J is any ideal in R containing I, then either J = R or J = IPrime ideals extend the idea of prime numbers to rings.  That is, rather than looking at properties of prime numbers, we can look at subsets of a ring that share many of the properties of prime numbers.  This enables us to look at mathematical structures other than the familiar ones that may share similar qualities.*

Modules over rings, for instance, are generalizations of vector spaces.  Vector spaces have scalars that form fields.  Modules only require scalars to form rings, which are simpler and may not have an identity element over module multiplication.  The primary difference between modules and vector spaces are that a module does not necessarily have a basis and certain modules may not have unique rank.

As with groups, homomorphisms are mappings from one ring to another that preserve structures and isomorphisms are homomorphisms that are also onto and 1-1.  As with the study of mappings on groups, the interest is in seeing what kinds of ring structures arise when we have mappings across rings that may not be entirely similar.

We will end the unit looking at a particular kind of ring, called the polynomial ring.  These are formed by polynomials in one or more variables with coefficients in some ring *
R*.  Again, the interest is in finding generalizations of what we know about familiar polynomials with number coefficients.  What we will learn is that polynomial rings are either prime or factor uniquely.

This unit will take you 42 hours to complete.

☐    Subunit 3.1: 10 hours

☐    Video: 3 hours

☐    Assignment: 2 hours

☐    Subunit 3.2: 5 hours

☐    Subunit 3.3: 5 hours

☐    Subunit 3.4: 4 hours

☐    Subunit 3.5: 5 hours

☐    Subunit 3.6: 3 hours

☐    Subunit 3.7: 3 hours

☐    Subunit 3.8: 1 hour

☐    Subunit 3.9: 5 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Define and generate rings.
• Determine whether or not a subring is an ideal.
• Determine whether or not a ring mapping is a homomorphism or isomorphism.

3.1 Definitions and Properties of Rings   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “16.1
Rings,”pages 239-244.

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• Lecture: Harvard University Extension: Dr. Benedict Gross’s “Math E-222 Abstract Algebra”: “Rings” Link: Harvard University Extension: Dr. Benedict Gross’s “Math E-222 Abstract Algebra”: “Rings Part 1,” "Rings Part 2," and "Rings Part 3" (YouTube)

Also Available in: Adobe Flash, Quicktime, or Mp3

Instructions: Please click on the links and watch each video in its entirety.  To choose another format, scroll down to Week 9 and choose the format most appropriate for your internet connection.  Then download Parts 1-3 of the “Rings: Examples of Rings and Basic Properties and Constructions” lectures listed in Week 9.  Please watch all of the videos in their entirety (about 53 minutes for Part 1, about 48 minutes for Part 2, and about 50 minutes for Part 3).  These video clips discuss properties and constructions of rings.

Terms of Use: This video has been posted with permission for non-profit educational use by Harvard University.  The original version can be found here.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problems 1 and 3” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications:“Rings”: “Exercise Problems 1 and 3” (PDF)

Also available in:

Instructions: Try to do problems 1 and 3 on page 257.  Then, check the solutions on page 405.

3.2 Example of Rings   - Reading: Wikipedia: “Ring (Mathematics)” Link: Wikipedia: “Ring (Mathematics)” (PDF)

Instructions: Please read the entire webpage.  This webpage contains a concise rendering of information on rings, including useful examples.

• Reading: Wolfram MathWorld: Eric W. Weisstein’s “Ring” Link: Wolfram MathWorld: Eric. W. Weisstein’s “Ring” (HTML)

Instructions: Please click on the link to read the information on the webpage for a concise rendering of information on rings.  It is short but has a number of links to related topics of interest.  Halfway down the page there is an interesting explanation of optional properties of rings and examples of rings that rise from changes in properties.

3.3 Commutative and Non-Commutative Rings   3.3.1 Commutative Rings   - Reading: Wikipedia: Commutative Ring Link: Wikipedia: “Commutative Ring” (PDF)

Instructions: Please read the entire webpage for a concise rendering of information on commutative rings, including various examples.  This topic is useful toward Abstract Algebra II.

3.3.2 Non-Commutative Rings   - Reading: Wikipedia: “Noncommutative Ring” Link: Wikipedia: “Noncommutative Ring” (PDF)

Instructions: Please read the entire webpage for a concise rendering of information on non-commutative rings, including various examples.  This topic is useful toward Abstract Algebra II.

3.4 Ideals   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “16.3 Ring
Homomorphisms and Ideals,” pages 246-250.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be
``````
• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 5” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings:” “Exercise Problem 5” (PDF)

Also available in:

Instructions: Practice what you have learned by completing problem 5 on page 258.  Then, check the solution on page 404.

3.5 Maximal and Prime Ideals   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “16.4
Maximal and Prime Ideals,” pages 250-252.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be
``````
• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 4” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 4” (PDF)

Also available in:

Instructions: Try to do problem 4 on page 258.  Then, check the solution on page 404.

3.6 Modules   - Reading: Wolfram MathWorld: Eric W. Weisstein’s “Module” Link: Wolfram MathWorld: Eric W. Weisstein’s “Module” (HTML)

Instructions: Please click on the link to read the information on the page.  This webpage contains a concise rendering of information on modules, which are abstractions of vector spaces, with the primary exception being the coefficients are over rings instead of fields.  An example is the set of integers, which is a module over itself.

Instructions: Please read the entire webpage for a concise rendering of information on modules, including various examples.

3.7 Ring Homomorphisms   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “16.3 Ring
Homomorphisms and Ideals,” pages 246-249.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be
``````
• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 19” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings”: “Exercise Problem 19” (PDF)

Also available in:

Instructions: Try to do problem 19 on page 259.  Then, check the solution on page 404.

3.8 Ring Isomorphisms   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Rings” (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “16.3 Ring
Homomorphisms and Ideals,” pages 249-250.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be
``````

3.9 Polynomial Rings   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Polynomials” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Polynomials” (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “Chapter
17: Polynomials,” pages 263-277.