Questions tagged [ring-theory]

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

A ring $R$ is a triple $(R,+,\cdot)$ where $R$ is a nonempty set such that $(R,+)$ forms an abelian group, $(R,\cdot)$ forms a semigroup, and the two operations are related by the distributive laws: $a\cdot(b+c)=a\cdot b+a\cdot c$ and $(b+c)\cdot a=b\cdot a+c\cdot a$.

Important examples of rings include domains (such as the integers), fields (such as the real numbers), square matrix rings, polynomial rings, and rings of functions. Rings are studied in their own right in abstract algebra, but they are also prominently used in number theory, geometry, algebraic geometry, and logic.

Many authors require the semigroup $(R,\cdot)$ to have an identity, often denoted $1_R$ or $1$. Many other authors do not make that requirement. This difference is something that students and posters should be aware of. Scholars of the former school call the structures not necessarily having a unit element . Scholars of the latter school call $R$ a ring with identity, when $1_R$ exists. This difference of opinions has an impact on the definition of a ring homomorphism. The scholars who include the presence of $1_R$ as an axiom assume that it is preserved under ring homomorphisms. The scholars who don't insist on the existence of $1_R$ obviously cannot make this requirement.

The operation $\cdot$ does not have to be commutative, but when it is, $R$ is called a commutative ring.

There are numerous types of rings studied in different ways. An ideal in a ring is the ring-theoretic analogue of a normal subgroup of a group. The study of ideals is an important component of ring theory.

This tag often goes along with the and/or tags.

19658 questions
48
votes
6 answers

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and let $A$ be a maximal ideal. Let $a,b\in R:ab\in A$. I'm trying to construct an ideal $B$ such that $A\subset B \neq A$…
Freeman
  • 4,739
  • 11
  • 47
  • 75
46
votes
3 answers

Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?

I know that $1$ is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=0$ is a prime ideal in $\mathbb Z$. Isn't $(p)$ being a…
Rasmus
  • 17,694
  • 2
  • 57
  • 91
46
votes
4 answers

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples. If one…
46
votes
8 answers

Is a ring a set?

We know that a ring consists of a set equipped with two binary operations. My question is whether a ring is a set or not. For example, we can have $(\mathbb{R},+,-)$ where $\mathbb{R}$ is a set and $+$ and $-$ are binary operations associated with…
Kun
  • 2,406
  • 3
  • 16
  • 26
46
votes
4 answers

A commutative ring is a field iff the only ideals are $(0)$ and $(1)$

Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$. I can't figure out where to start other that I need to prove some biconditional statement. Any help?
Jackson Hart
  • 1,482
  • 2
  • 16
  • 31
45
votes
7 answers

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems.…
45
votes
1 answer

Is Tolkien's Middle Earth flat?

In the first introductory chapter of his book Gravitation and cosmology: principles and applications of the general theory of relativity Steven Weinberg discusses the origin of non-euclidean geometries and the "inner properties" of surfaces. He…
xaxa
  • 880
  • 5
  • 12
43
votes
1 answer

Submodule of free module over a p.i.d. is free even when the module is not finitely generated?

I have heard that any submodule of a free module over a p.i.d. is free. I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, so it does not apply to modules that are not…
41
votes
2 answers

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ring with no maximal ideals. A homework question in…
40
votes
4 answers

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do not have unique factorization, and in fact Kummer…
40
votes
5 answers

Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that $ba_0=ba_1=\cdots=ba_n=0$?
Mohan
  • 13,820
  • 17
  • 72
  • 125
40
votes
2 answers

How to deal with polynomial quotient rings

The question is quite general and looks to explore the properties of quotient rings of the form $$\mathbb{Z}_{m}[X] / (f(x)) \quad \text{and} \quad \mathbb{R}[X]/(f(x))$$ where $m \in \mathbb{N}$ Classic examples of how one can treat such rings is…
Mathmo
  • 4,697
  • 4
  • 37
  • 71
39
votes
8 answers

Why is $\mathbb{Z}[X]$ not a euclidean domain? What goes wrong with the degree function?

I know that $K[X]$ is a Euclidean domain but $\mathbb{Z}[X]$ is not. I understand this, if I consider the ideal $\langle X,2 \rangle$ which is a principal ideal of $K[X]$ but not of $\mathbb{Z}[X]$. So $\mathbb{Z}[X]$ isn't a principal ideal domain…
Marc
  • 3,057
  • 2
  • 17
  • 27
38
votes
6 answers

Prove that the Gaussian Integer's ring is a Euclidean domain

I'm having some trouble proving that the Gaussian Integer's ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i've got so far. To be a Euclidean domain means that there is a defined application (often called norm) that verifies this two…
38
votes
2 answers

What are the left and right ideals of matrix ring? How about the two sided ideals?

What are the left and right ideals of matrix ring? How about the two sided ideals?
user8195