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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3
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2answers
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An exercise about zerodivisors

If $A$ is a commutative ring with unity, $f\in A$ and $x\in SpecA$, with the notation $f(x)$ I mean the coset $x+f\in A/x$. Now look at this exercise: Prove that a nonzero element $f\in A$ is a ...
2
votes
0answers
43 views

Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
6
votes
1answer
48 views

Unity in the rings of matrices

Suppose we are given an arbitrary ring $R$. Then the set $M_n(R)$ of all square matrices with elements from $R$, together with usual matrix addition and multiplication forms a ring. If R is a unitary ...
3
votes
2answers
506 views

How can a finite integral domain have a non-zero characteristic?

I know that by virtue of being a finite integral domain, $n\cdot 1_{R}=m\cdot 1_{R}$. Hence, $(n-m)\cdot 1_{R}=0$. The smallest positive element in the set of all possible such $(n-m)$'s is the ...
3
votes
1answer
106 views

Can a finitely generated $\mathbb{Z}$-algebra contain $\mathbb{Q}$?

Is there a ring between $\mathbb{Q}$ and $\mathbb{R}$ that is finitely generated as an algebra over $\mathbb{Z}$? My guess is there isn't. I can see that it would have to be finitely generated over ...
2
votes
1answer
47 views

In the lattice of ideals, what are the lowerbounds of the prime ideals?

Take the lattice of ideals in a non-commutative ring with 1. It is well known that the lowerbounds of all the maximal ideals are the superfluous ideals. Is there a similar characterization for the ...
5
votes
2answers
355 views

Prove that semi-simple rings are Dedekind-finite

Just to be consistent with the terminology, let me define the words I'm using. A module $M$ is simple if it has no proper non-zero submodules, and semi-simple if it can be written as a direct sum $M ...
1
vote
1answer
79 views

Prime and semiprime ideals of $A=T_3(D)$, the ring of $3\times 3$ upper triangular matrices over $D$

Let $D$ be a division ring. Could anyone tell me which are the prime and semiprime ideals of $A=T_{3}(D)$, where $A=T_{3}(D)$ the ring of $3\times 3$ upper triangular matrices with coefficients in ...
3
votes
1answer
125 views

Prime and Primary Ideals in Completion of a ring

Let $(R,\mathfrak m)$ be a local noetherian ring and $\widehat{R}$ its $\mathfrak m$-adic completion. If $\mathfrak q\in \operatorname{Spec}(\widehat{R})$ then can we find $\mathfrak p\in ...
2
votes
3answers
169 views

What's the motivation of the ideal? [duplicate]

I'm reading a book on Algebra, it introduces the concept of ring after some examples, the concept of ideal. Definition I.1.8. Let $(A,+,\cdot)$ be a ring and $I$ a non-empty subset of $A$. We say ...
2
votes
1answer
104 views

If $V$ is a vector space over a division ring $K$, and $A=\mathrm{End}_K(V)$, then every quotient ring of $A$ is a prime ring

Let $K$ be a division ring, let $V=V_{K}$ a vector space over $K$, and let $A=\mathrm{End}_{K}(V)$. Could anyone give me an idea of ​​how to prove that every quotient ring of $A$ is a prime ring?
6
votes
1answer
111 views

“Almost” ring homomorphism

This is an exercise out of Herstein which seems pretty straightforward but is eluding me. Let $R,R'$ be rings and let $\phi:R\to R'$ be a mapping such that, for every $x,y\in R$: $${\rm ...
4
votes
4answers
177 views

Proof of the uniqueness of maximal ideal

Let $R$ be a commutative ring with $1$. Let $M$ be a maximal ideal of $R$ such that $M^2 = 0$. Prove that $M$ is the only maximal ideal of $R$.
1
vote
1answer
95 views

Problem to understand Hungerford's book

In the Hungerford's book and following the answers of this question: Help to understand the ring of polynomials terminology in $n$ indeterminates I have troubles to understand the following remark in ...
7
votes
3answers
204 views

Ring Inside an Algebraic Field Extension

Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?
2
votes
2answers
101 views

Help to understand the ring of polynomials terminology in $n$ indeterminates

In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner: After the author defines the operations in this ring with a theorem: ...
2
votes
1answer
303 views

If $a^3=a$ in a ring, prove: the ring is commutative [duplicate]

Let $R$ be a ring, not necessarily with a unit element. $R$ is not necessarily integral. If for any $a \in R$, $a^3=a$, prove: $R$ is commutative: Any $a, b \in R$, $ab=ba$. My efforts on it: I can ...
4
votes
4answers
337 views

The elements of $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$

I'm really confused with this one... How can I determine the elements of the module $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$? Or its cardinality? Does ...
-4
votes
2answers
107 views

Help with Theorem III.3.11 in Hungerford's algebra book

I need help to prove part (i) of this theorem which I couldn't prove. Any help would be appreciated. Thanks in advance.
4
votes
3answers
66 views

Factorize in R[x]

I have the polynomial $x^8+1$, I know that there's no root for solve this in $\Bbb R[x]$ but i want to factorize this to the minimal expression. This is possible or this is irreducible?
2
votes
2answers
101 views

Why the terms “unit” and “irreducible”?

I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition Maybe historical reasons? For example, I suppose the second ...
3
votes
2answers
106 views

Exercise about prime ideals

Let $A$ be a ring. Prove that the following conditions are equivalent: $i)$ All ideals $I \subsetneq A$ are prime. $ii)$ The set of all ideals of $A$ is totally ordered by inclusion and all ideals ...
4
votes
1answer
96 views

The Baer-McCoy (a.k.a. prime) radical of $A$

Let $B$ a ring and let $A$ a subring of $B$. Show that $A\cap \mathrm{Nil}_{*}(B)\subset \mathrm{Nil}_{*}(A)$. If $A$ is contained in the center of $B$, show that $A\cap ...
0
votes
1answer
74 views

Questions on (subring/ submodule) of a graded (ring/ module)

I have a question which seems a bit silly... If we have $R$ a graded ring, does it follow that every subring of $R$ is also graded? Because I have a problem here as such: I have a graded ring $R$ ...
2
votes
0answers
47 views

Find the factorization of the polynomial as a product of irreducible [duplicate]

Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$ Testing with the simplest possible root in this case, $P(1)=0$ Applying the ...
1
vote
1answer
100 views

Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]

Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$ Testing with the simplest possible root in this case, $P(1) = 0$ ...
7
votes
1answer
190 views

Calculating in quotient ring of $\mathbb{R}[X]$

Part of an old Oxford exam (1992 A1) We want to find which elements of the quotient ring $\mathbb{R}[X]/(x^3-x^2+x-1)$ are equal to their own square. Now, we note first that ...
5
votes
1answer
91 views

Generalising the Chinese Remainder Theorem

We have that for $I,J$ ideals of some ring $R$ with $R=I+J$, $$\frac{R}{I\cap J} \cong \frac{R}{I} \times \frac{R}{J}$$ My question is whether the analogous expression for three ideals $I,J,K$ where ...
5
votes
1answer
83 views

Express $4+\sqrt{-2}$ as a product of irreducibles

This is part of an old Oxford Part A exam paper. (1992 A1) Suppose we equip $R=\mathbb{Z}[\sqrt{-2}]$ with the Euclidean function $d$ defined by $$d(m+n\sqrt{-2})=|m+n\sqrt{-2}|^2$$ I want to ...
1
vote
4answers
188 views

Subrings of $\mathbb{Q}$

Let $p$ be prime. Suppose $R$ is the set of all rational numbers of the form $\frac{m}{n}$ where $m,n$ are integers and $p$ does not divide $n$. Clearly then $R$ is a subring of $\mathbb{Q}$. I now ...
4
votes
3answers
281 views

example of a flat but not faithfully flat ring extension

I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
6
votes
4answers
843 views

Quotient rings of Gaussian integers $\mathbb{Z}[i]/(2)$, $\mathbb{Z}[i]/(3)$, $\mathbb{Z}[i]/(5)$,

I am studying for an algebra qualifying exam and came across the following problem. Let $R$ be the ring of Gaussian Integers. Of the three quotient rings $$R/(2),\quad R/(3),\quad R/(5),$$ one ...
10
votes
3answers
471 views

Irreducibility of $x^n-x-1$ over $\mathbb Q$

I want to prove that $p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible. My attempt. GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
4
votes
1answer
199 views

How to compute Nakayama functor explicitly?

I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
2
votes
1answer
186 views

Do there exist polynomial rings with nonzero prime ideals that are not maximal?

I know that with $F$ a field, $F[x]/(f(x))$ is a field iff $f(x)$ is irreducible in $F$. Due to the fact that in a UFD irreducible elements are necessarily prime, we would have that $(f(x))$ is both ...
0
votes
2answers
277 views

Rings | Homomorphisms | Units

Question Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$. Attempt ...
2
votes
2answers
541 views

Under what conditions is a zero divisor element $a$ in commutative ring $R$ nilpotent?

Suppose that $R$ is a commutative ring with identity $1$ Let $a\in R$ with $ab=0$ for some $b\ne0$. Under what conditions $a$ must be also nilpotent?
2
votes
2answers
64 views

length of sum of two submodule

Let $M$ be a $R$-module with finite length and $K$ and $N$ be a submodule of $M$. Prove that $l(K+N)+l(K\cap N)=l(K)+l(N)$. My proof: First, by assuming that $K\cap N=\{0\}$, we can conclude that ...
6
votes
1answer
80 views

Given $G$, when can we find a division ring $R$ with $R^*=G$?

This is motivated by a characterization of finite cyclic groups, in which one proves Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic. The proof is ...
1
vote
1answer
94 views

A question in Ring theory [duplicate]

R be a commutative ring with unity and it has exactly one maximal ideal. Then prove that the equation $x^2 =x$ has exactly two solutions. Show me the right way to solbe this one.
1
vote
1answer
110 views

Finite ring of sets

I have some questions about finite rings of sets and I'll be very grateful for any help. Let E be some fixed non-empty set. Suppose we are given some finite ring of subsets of set E, i.e. some ...
8
votes
1answer
118 views

Amenable group rings embeddable in skew fields

I'm looking for a reference of the following fact: given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent: (1) the group ring $K[G]$ is a domain; (2) $K[G]$ is ...
9
votes
2answers
185 views

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions. I know that $0$ and $1$ are the solutions, but I can't proceed ...
4
votes
2answers
113 views

Is the kernel of a homomorphism from a Boolean ring to $\mathbb{Z}_2$ always a maximal ideal?

Let $(B, +, \cdot)$ be a ring (not necessarily unital!) with the property that every $x \in B$ satisfies $x \cdot x = x$. How does one show that the kernel of any non-zero homomorphism of rings ...
4
votes
1answer
91 views

Number of left ideals in a simple ring

I'm puzzling over a few algebra questions: 1) Give an example of a simple ring with exactly $12$ non-zero proper left ideals. For this one I have no idea, I am not good with coming up with examples ...
7
votes
1answer
137 views

Artinian rings are perfect

Definition. A ring is called perfect if every flat module is projective. Is there a simple way to prove that an Artinian ring is perfect (in the commutative case)?
8
votes
2answers
91 views

Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?

Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
22
votes
4answers
678 views

Why do we study prime ideals?

I hope this isn't an inappropriate question here! I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
3
votes
2answers
197 views

A noetherian ring whose ideals are idempotent is artinian

I have to prove the folowing: If $R$ is a Noetherian ring, and for every ideal $I$ of $R$ we have $I = I^{2}$, then $R$ is Artinian. My first thought was to try to prove that the nilradical of ...
3
votes
1answer
141 views

Is the endomorphism ring of $\mathbb{R}$ self-opposite?

Is the endomorphism ring $End\mathbb{R}$ of the Abelian group $\mathbb{R}$ isomorphic to its opposite ring? All subrings of a self-opposite ring are self-opposite. By choosing an isomorphism of ...