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

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4
votes
4answers
564 views

If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$?

Let $I = \langle 2\rangle$. Prove $I[x]$ is not a maximal ideal of $\mathbb Z[x]$ even though $I$ is a maximal ideal of $\mathbb Z$. My professor mentioned that I should try adding something to ...
9
votes
3answers
484 views

Coproduct in the category of (noncommutative) associative algebras

For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras ...
5
votes
3answers
158 views

Can the product of non-zero ideals in a unital ring be zero?

Let $R$ be a ring with unity and $0\neq I,J\lhd R.$ Can it be that $IJ=0?$ It is possible in rings without unity. Let $A$ be a nontrivial abelian group made a ring by defining a zero multiplication ...
6
votes
2answers
380 views

Artinian if and only if Noetherian

Let $R$ be a ring (commutative, with identity), $m$ a maximal ideal and $M$ an $R$-module. Suppose $m^nM=0$ for some $n>0$. Then $M$ is Noetherian if and only if $M$ is Artinian Do you have any ...
2
votes
1answer
187 views

Subalgebra of free associative commutative algebra which is not free

I'm trying to provide a counterexample for analogue of Nielsen–Schreier theorem for the variety of associative commutative algebras (not necessary with unity) over a filed $F$. A counterexample for ...
2
votes
3answers
619 views

The ring $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$

The set $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ spans a ring under real addition and multiplication. Which elements have multiplicative inverses? This is part of an exercise from an introductory text ...
4
votes
2answers
134 views

An equivalent condition for an element to be integral

Let $R$ be a noetherian domain, $Q$ its field of fraction and $u\in Q$. Could you help me to prove that $u$ is integral over $R$ if and only if there exists $r\in R$ $r\neq0$ and $ru^n\in R$ for ...
2
votes
0answers
89 views

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent?

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent? Particularly I'd like to know the formulation thereof which concerns the kernel of a surjective ring ...
1
vote
1answer
50 views

How to make $R:=\{ \sum_{i=0}^{\infty} c_i A^i$ with only finitely many $c_i \neq 0 \}$ a commutative ring with one?

Let $A$ be an $n$ by $n$ matrix over a field $K$, define $R:=\{ \sum_{i=0}^{\infty} c_i A^i$ with only finitely many $c_i \neq 0 \}$ Could anyone help me show you can turn $R$ into a commutative ring ...
2
votes
1answer
118 views

Why is this homomorphism an isomorphism?

Let $R$ be a commutative ring with identity. Suppose $R=(r_1,\ldots,r_k)$. Take an homomorphism of $R$-modules: $f:M\rightarrow N$. Suppose that the function $\frac{f}{1}:M_{r_i}\rightarrow N_{r_i}$ ...
2
votes
3answers
501 views

If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal

If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal I have the ideal test but no clue as to what to do with it: $a-b \in A$ whenever $a,b \in A$ ra and ar are ...
3
votes
5answers
242 views

units in commutative rings

An element $a$ of a ring $R$ is called a unit if it has a two sided inverse under multiplication; that is, if there exists $b \in R$ such with $ab = ba = 1_R$. How would you show that if $R$ is ...
11
votes
2answers
423 views

Showing a UFD which is not a PID must have a nonprincipal maximal ideal.

Given that $R$ is a UFD which is not a PID, I want to show that $R$ must have a nonprincipal maximal ideal. I tried several methods, including Zorn's lemma but didn't get anywhere. Any suggestions ...
3
votes
3answers
248 views

Why is this ring semisimple?

Let $R$ be a simple ring (i.e. a ring with no nontrivial two-sided ideals) which contains a left ideal which is simple as a left $R$-module. How can I prove that $R$ is semisimple?
3
votes
1answer
166 views

If an ideal $I$ contains a non-zero-divisor then $\mathrm{End}_R(I)$ is commutative

How can I prove that if $I$ is an ideal of a commutative ring $R$ that contains a non-zero-divisor then $\mathrm{End}_R(I)$ is commutative?
8
votes
1answer
229 views

Ties between Lie algebras and ring theory

I would like to get a general understanding of the relationship between (noncommutative) ring theory and Lie algebra theory. All Lie algebras are finite dimensional and over a field $k$ of ...
2
votes
3answers
273 views

Is $AxA$ a two-sided ideal for an element $x$ of a ring $A$?

In Bourbaki's Algebra there is the following proposition: Let $A$ be a ring (with $1$), $(x_\lambda)_{\lambda\in L}$ a family of elements of $A$ and $\mathfrak{a}$ the set of sums $\sum_{\lambda\in ...
0
votes
2answers
194 views

Is a ring closed under both operations?

A ring is a set R, together with two binary operations $+, \cdot : R\times R \to R$ that satisfy $(R,+)$ is an abelian group Associativity Distributivity Multiplicative identity so $\exists 1_R ...
3
votes
1answer
574 views

Zero divisor, nilpotent elements of quotient ring $\mathbb Z_2[x] / \langle x^8-1\rangle$

Consider the ring $R = \displaystyle\frac {\mathbb Z_2[x]}{\langle x^8-1\rangle}$. i) Is $R$ a finite ring? ii) Does $R$ have a zero divisor? iii) Does $R$ have nilpotent elements? ...
7
votes
3answers
6k views

what is difference between a ring and a field

The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra ...
1
vote
1answer
210 views

If $R$ is a finite dimensional algebra over a field then $R$ is simple as a ring if and only if it has a faithful simple left $R$-module

How can I prove that if $R$ is a finite dimensional algebra over a field then $R$ is simple as a ring if and only if it has a faithful simple left $R$-module?
4
votes
2answers
315 views

Showing polynomials in $k[x_1, \ldots , x_n]$ are irreducible

It is often the case when I wish to show a particular polynomial in $k[x_1, \ldots ,x_n]$ is irreducible. Assuming that the polynomial is sufficiently friendly (i.e. one I would encounter as part of a ...
4
votes
3answers
312 views

Rings with isomorphic proper subrings

Rings will be unital here but I don't require that subrings share the identity elements with superrings. I just accidentally came up with an example of a ring $R$ with a proper subring $S$ such that ...
3
votes
2answers
173 views

Surjective graded homomorphism of rings also an isomorphism?

Suppose we are given two graded (commutative) rings $A$ and $B$ and a graded homomorphism $\psi:A\rightarrow{B}$ between them. Suppose moreover that $\psi$ is surjective in each degree i.e. that ...
5
votes
3answers
144 views

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
2
votes
1answer
109 views

Associative division subalgebras of split Cayley-Dickson algebra

Let's consider the split Cayley-Dickson algebra $C$ over an arbitrary field $F$ (It is well known that all split composition algebras having the same dimension over base field are isomorphic, e.g., ...
6
votes
0answers
123 views

On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.

I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
2
votes
2answers
190 views

What if $\operatorname{char}\mathbb{K}$ is not $0$ or if $\mathbb{K}$ is not algebraically closed? (Nullstellensatz)

Given a field $\mathbb{K}$ which is algebraically closed and of characteristic 0, we can say exactly what the maximal ideals of $\mathbb{K}[x_1,\dots,x_n]$ are and they correspond to points in ...
2
votes
0answers
65 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
2
votes
1answer
100 views

Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$

Prove: Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$. Note: In my book it defines $E_g\colon K[x] \to K[x]$ by sending $x$ to $g$. This seems like it ...
4
votes
3answers
491 views

The intersection of distinct maximal ideals is not prime.

Let $P,Q$ be distinct maximal ideals of a ring $R$. Prove that $P\cap Q$ is not prime. I am not sure how to prove this. The only facts that I can think of applying are the definitions, $R/M$ is ...
4
votes
3answers
1k views

Left inverse implies right inverse in a finite ring

Let $R$ be a finite ring, and assume $\exists x,y\in R$ such that $ xy=1$. How can I show it implies $yx=1$?
0
votes
2answers
263 views

Sided inverses in a non-commutative ring

I've asked myself the following question : does there exist a non-commutative ring $R$ with unity $1$ and elements $x,y,z \in R$ such that $xyz = 1$ but $y$ has no left nor right inverses? (Perhaps I ...
3
votes
3answers
118 views

Are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules all isomorphic to one another?

After looking back over some finite field theory, I've been thinking about the ring $\mathbb{Z}/p^k\mathbb{Z}$ for some prime $p$. I'm just curious, are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules ...
6
votes
1answer
238 views

Is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$?

If $p\in\mathbb{N}$ is a prime, is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$? I've proved that any non-unit factor in $\mathbb{Z}[x]$ must have degree at least 2. Eisenstein's criterion doesn't ...
0
votes
1answer
152 views

What is the relation between prime ring and indecomposable ring?

Prime ring means $0$ is prime in the ring, so $R$ is a prime ring iff for any two elements $a$ and $b$ of $R$, $arb = 0$ for all $r\in R$ implies that either $a = 0$ or $b = 0$. Indecomposable ring ...
15
votes
2answers
1k views

A subring of the field of fractions of a PID is a PID as well.

Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID? Any ...
1
vote
2answers
257 views

Followup to “Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$”

In this post: Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$ a nice example was given of a non-distributive ring. The lattice of ideals turned out to be the Diamond lattice $M_3$ with ...
1
vote
0answers
111 views

Unit group of quotient of noncommutative polynomial ring

In this recent post the original question led people to look for rigid, noncommutative rings. (Rigid means that the only endomorphisms are zero and the identity). Several (somewhat complicated) ...
0
votes
1answer
132 views

Prove that $[R:I] =\{r \in R\mid xr \in I\text{ for every }x \in R\}$ is an ideal of $R$ that contains $I$

If $I$ is an ideal in a ring $R$ let $[R:I] =\{r \in R\mid xr \in I\text{ for every }x \in R\}$. How can I show that $[R:I]$ is an ideal of $R$ which contains $I$.
5
votes
2answers
326 views

certain examples of fields of fractions

Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of ...
5
votes
1answer
354 views

Ideals in non-associative rings and the identity $(xy)z=y(zx)$.

I have come across this paper. The authors prove that magmas satisfying the identity $$(xy)z=y(zx)\tag1$$ are nearly both associative and commutative. To be precise, they show that in such magmas, ...
14
votes
1answer
270 views

Ideal in an Artinian Ring $I=aR=Rb$, prove $I=Ra=bR$

Let $R$ be an Artinian Ring and suppose there exists $a,b\in R$ s.t. $I=aR=Rb$, then prove $I=bR=Ra$. (You may assume that a right Artinian Ring is Right Noetherian). I've managed to get $Ra$, ...
1
vote
0answers
45 views

$GL_2$-Invariants of $\mathbb{C}[X,Y]$

One of the problems in some work I'm doing tells me to consider $GL_2$ acting on $\mathbb{C}[X,Y]$, induced by the natural representation of $GL_2$ on $\mathbb{C}^2$. I just wanted to check 2 things: ...
2
votes
2answers
410 views

How to show that if two integral domains are isomorphic, then their corresponding field of quotients are isomorphic?

If two integral domains $D$ and $D'$ are isomorphic show that their corresponding field of quotients (fractions) $Q(D)$ and $Q(D')$ are isomorphic.
2
votes
1answer
101 views

Is $\bigwedge(V)$ self-injective?

For a vector space $V$, is the grassman algebra $\bigwedge(V)$ always an injective module over itself? Is there a proof, or even just a brief explanation?
1
vote
0answers
99 views

Question on Bergman's Diamond Lemma.

Last week I was recommended Bergman's Diamond Lemma in these comments. I read through the paper, and was working on an exercise in it. I want to know if the reduction systems $\{(x^2y^2,yx)\}$ and ...
6
votes
2answers
205 views

An $R$ module and $S$ module that cannot be an $R$-$S$ bimodule

In connection with this question: Modules and tensor products Question: For two commutative rings $R$ and $S$ (with unity), is there an abelian group $M$ which has $R$ module and $S$ module ...
1
vote
1answer
187 views

Example request: simple, radical ring

I'm looking for an example of a ring $R$ (necessarily nonunital) which is simple (in the sense that $R \cdot R \neq 0$ and $R$ has no proper, nonzero 2-sided ideals) and also radical (in the sense ...
7
votes
3answers
353 views

Module M/IM of finite length $\implies$ Ring A/I of finite length

This question is due to a proof in an algebra book (on the topic of dimension theory) which I don't fully understand (specifically, the proof of Thm 6.9b) in Kommutative Algebra by Ischebeck). It may ...