Why Are Rings in Abstract Algebra Important?

[dionysian]

Ok so I am not a math major and i haven't taken an abstract algebra class but i am curoius about the subject. I have been watching video lectures at UCCS at http://cmes.uccs.edu/Fall2007/Math414/archive.php?type=valid and the proffessor talks about groups and rings. In the introduction the groups he states that the definitions of a group are sufficient for solving a linear equation. He states that this is the intuition beheind the idea of a group. Now when he introduces a ring he doesn't seem to give much motivation beheind it.

My questions is this: Why are rings important? Is there any intuition behind them that warrent studying them and/or giving them there own name?

 

[rasmhop]

Because groups may be appropriate for studying the solutions of equations of the form ax=b, but to study ax+by=c we need rings. A group is a set with a group structure. A ring is a set with two structures: one abelian group structure whose operation we denote by +, and one monoid structure whose operation we denote by *, and these operations are connected by distributivity. The standard answer is that rings is the appropriate way to generalize the integers.

For an example of why rings are important consider polynomials. The set of monomials can be considered as a monoid. Actually the set of monic monomials in n variables with coefficients in R is actually the free abelian monoid in n letters, and the set of monomials is the product monoid A x M where A is the monoid of coefficients and M is the monic monomials. To extend this to polynomials we need to consider sums of monomials, and therefore to introduce the polynomial ring.

Also if you have taken linear algebra and understand the motivation of a field, remember that a field is just a generalization of a ring. A field is just a ring whose multiplicative structure is actually an abelian group. So a field is a ring in which every non-zero element x has a multiplicative inverse y (s.t. xy=1) and where multiplication is commutative.

For a somewhat more advanced discussion of this question see "http://mathoverflow.net/questions/2748/what-is-the-right-definition-of-a-ring" " at mathoverflow.

 

[rochfor1]

I like to think of abstract algebraic structures in terms of what algebraic aspects of matrices they model (I may or may not be a functional analyst...)

Groups capture the idea of multiplication in the set of invertible matrices.
Rings capture the interaction between matrix multiplication and addition.
Modules capture the interaction between matrix addition and scalar multiplication.
Algebras capture the interaction of matrix multiplication and addition and scalar multiplication.

There certainly are many other examples of all these abstract algebraic structures that are not matrices, but the success of various representation theories shows that matrices are very important examples of each.

 

[wofsy]

rings arise whenever you have both addition and multiplication on the same space - where addition is commutative. To me this is more intuitive than just a group where there is only one law of multiplication. The most natural ring is the integers but the number of different rings and applications of them is boundless.

Rings of polynomials and their quotients are huge - as are rungs of functions.