Mathematics Stack Exchange
Mathematics Stack Exchange

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: $$a\cdot(b+c)=a\cdot b+a\cdot c\quad\text{and}\quad(b+c)\cdot a=b\cdot a+c\cdot a$$

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.

21353 questions
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$Ra=Rb$ if and only if $aR=bR$

On which classes of (non commutative) rings we have the following property: $aR=bR$ if and only if $Ra=Rb$ ? While I googling around I found the notion of "Duo Ring" in which $aR=Ra$ for every $a\in R$. This is stronger that what I am looking for.…
user9077
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The subring test (subtraction vs. addition closure)

This is how the Wikipedia article on subring defines the subring test The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and multiplication, and contains the multiplicative identity…
Florian J
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subideals of an ideal

If $R$ is a ring, J is an ideal in $R$, and $I$ is an ideal of $J$ (with $J$ considered as a ring), does it follow that $I$ is an ideal of $R$? That is, is $I$ necessarily closed under multiplication by elements of $R$? Surely $I$ is closed under…
Howard Shaw
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Rings Isomorphisms

I study ring theory, and I wondered about the next question that I asked myself for practice: Let $\mathcal{R}_1, \mathcal{R}_2$ be ring, and let $G_1, G_2$ be their additive groups, and $M_1, M_2$ be their multiplicative monoids. Suppose there are…
Or Shahar
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Clearly, $3 \in \mathbb{Z}$ is not a unit, because $1/3 \notin \mathbb{Z}$. What theorem does this kind of reasoning appeal to?

Intuitively, we may conclude that $3$ is not a unit in $\mathbb{Z}$ simply by observing that $1/3 \notin \mathbb{Z}$. However, what does this reasoning actually appeal to??? Is it true that: Conjecture. For all rings $R$ and all $r \in R$, if there…
goblin GONE
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Field Extension Notation

I've seen similar questions asked here, but I've not been able to find a comprehensive answer. I know that for a ring $R$, $R[X]$ denotes the ring of polynomials over $R$ and $R(X)$ denotes the field of fractions of $R[X]$. But if $\alpha \in S$,…
BusySignal
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How to show associativity of multiplication of polynomials in $R[x]$, where $R$ is a commutative, associative ring

Suppose that we have a commutative, associative ring $R$ which we use to generate the polynomial $R[x]$. Then $p(x) \in R[x]$ is of the form $p(x) = \sum_{i=0}^n a_i x^i$ for some $n \in \mathbb{N}$. Now, I wish to understand why $R[x]$ is itself a…
Harry Williams
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Show that $\left(1+\dfrac{x}{n}\right)^n \to e^x$ as $n \to \infty$ in a normed ring $R$

I had an interesting discussion yesterday with one of my friends (I think he is a member here, am I right?). He claimed that $$\left(1+\dfrac{x}{n}\right)^n \to e^x$$ basically in any normed ring $R$ (with a copy of $\mathbb{Q}$, am I right?) as $$n…
Anonymous - a group
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Sum of ideals in a ring

I have the following question: If $I,J$ are ideals in $R$, then denote $I+J = \{ r \in R | r = x+y, x \in I, y \in J \}$ It is not hard to show that $I+J$ is again an ideal in $R$ If we then assume that $I+J = R$ we have the identity $R/(I \cap J)…
Juan S
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10
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Does a homomorphism from a unital ring to an integral domain force a multiplicative identity?

This is a question in Herstein's Topics in Algebra ("unit element" refers to multiplicative identity): If $R$ is a ring with unit element $1$, and $\phi$ is a homomorphism of $R$ into an integral domain $R'$ such that $\ker\phi\ne R$, prove that…
Matthew Kwan
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Is there a ring so that any two distinct non-zero elements do not commute?

I was wondering if there was a ring so that any two distinct non-zero elements do not commute. Formally, is there a ring $R\not=\{0\}$ so that $$\forall x,y\in R\setminus\{0\}, x\not= y\implies xy\not=yx$$ Obviously, $\{0\}$ and $\Bbb Z/2\Bbb Z$…
xavierm02
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Show that the ideal $ (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} [\sqrt{-7} ] $ is not principal

Show that the ideal $ I = (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} [\sqrt{-7} ] $ is not principal. My thoughts so far: Work by contradiction. Assume that $ I $ is principal, i.e. that it is generated by some element $ z = a + b\sqrt{-7} \in…
user938272
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Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent iff $a_0,a_1,a_2,\ldots,a_n$ are nilpotent

Let $R$ be a commutative ring. Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent if and only if $a_0,a_1,a_2,\ldots,a_n$ are nilpotent. Now since $f(x)$ is nilpotent then $f^n=0$ for some $n$. Now $f^2=c_0+c_1 x+\cdots+c_n x^n$…
Learnmore
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Are there fields $F$ such that the rings $F[x]/(x^2)$ and $F[x]/(x^2-1)$ are isomorphic?

My first instinct is to go to the 1st isomorphism theorem and say that the two rings are isomorphic if there is a field where $(x^2)=(x^2-1)$, which doesn't seem terribly promising. But then it seems to me that $F[x]/(x^2)$ is the ring you get when…
jobrien929
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Is this ring a well known ring and if so how is it called?

I just had this thought when I was thinking how I was introduced to the concept of number in primary school and I came upon the conclusion that the numbers we were taught to manipulate (adding, multiplying, long division), i.e. the rational numbers…
Marc Bogaerts
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