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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2
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2answers
121 views

Let $(R,m)$ be a Noetherian local commutative ring. And suppose that $m^{n} = 0$ for some $n \in \mathbb{N}$

Let $(R,m)$ be a Noetherian local commutative ring. And suppose that $m^{n} = 0$ for some $n \in \mathbb{N}$ Then I want to show that $m/m^2$ is a finite dimensional vector space over the field ...
2
votes
1answer
416 views

Is every injective rational function $f:\mathbb Q\to\mathbb Q$ a polynomial?

I thought this might be quite easy to show, and then realized that the tools I know from real analysis aren't going to help here. Suppose we have a rational function: $$ f(X)=\frac{P(X)}{Q(X)} $$ ...
0
votes
1answer
66 views

Symmetric difference or addition on power set?

In my text book, it shows that $3\{1,2\}=\{1,2\}+\{1,2\}+\{1,2\}$ $=(\{1,2\}+\{1,2\})+\{1,2\}$ $= \emptyset +\{1,2\}$ $=\{1,2\}$ Here, I don't understand how $(\{1,2\}+\{1,2\})$ became ...
1
vote
3answers
79 views

$a$ is a prime and $R$ is an integral domain $\implies a$ is irreducible

I noticed the following statement in a proof of a theorem on Euclidean rings. $a$ is a prime and $R$ is an integral domain $\implies a$ is irreducible It seems like this part of the statement is ...
1
vote
1answer
118 views

Find solutions to equation (ring/field theory, residue class)

I'm trying to solve this problem: A residue class ring mod $n$ is a field if n is prime. Let $\mathbb{Z}_p$ be a residue ring, p prime. Let $a \in \mathbb{Z}_p$. What are all solutions $x \in ...
1
vote
1answer
46 views

Necessary and sufficient conditions for an operation to be the multiplication operation of a ring

Let $S$ be a set with at least two elements $1$ and $0$, which are distinct. Suppose $*$ is an operation on $S$ such that $*$ is associative, $x*0=0*x=0$ for all $x$, and $x*1=1*x=x$ for all $x$. ...
1
vote
1answer
65 views

Power set-Commutative ring

$\mathbb{Z},\mathbb{Q}$... are commutative rings because, for example, $xy=yx, x, y\in \mathbb{Z}$? Could you give me examples for power set being commutative ring?
4
votes
1answer
70 views

$N(\alpha) = \pm 1 \implies \alpha$ is invertible - Is my proof correct?

Let $\mathbb{Z}[\sqrt{2}] = \{a + b\sqrt{2}\mid a, b \in \mathbb{Z}\}$ Let $N(a + b\sqrt{2}) = a^2 - 2b^2$ Let $N(\alpha\beta) = N(\alpha)N(\beta)$ for $\alpha, \ \beta \in \mathbb{Z}[\sqrt{2}]$ ...
3
votes
1answer
188 views

A question in ring

I am revising for the finals, these questions on book I just cannot answer. Hope somebody can help me. Thanks Suppose that a belongs to a ring and $a^4=a^2$. Prove that $a^{2n}=a^2$for all ...
2
votes
1answer
67 views

Show that in a ring with $u^2 = 0$, $1 + u$ is a unit

I eventually worked this out through trial and error and found that $(1 + u)(1 - u) = 1$ But is there a more calculated way of determining this? I just kept trying different values until I found one ...
1
vote
1answer
76 views

Decomposition of a non chain ring

I'm dealing with decomposition of the quotient ring ${{\mathbb{Z}_2}\left[ {x,y} \right]}$ over ${\left( {{x^3} - 1,{y^3} - 1} \right)}$. I know that I should use the Chinese remainder theorem but I ...
2
votes
0answers
30 views

“Rule's” for reducilibility depending on degree of a polynomial

I want to make sure I have the following information correct. Here is what I understand regarding the reducibility of polynomials of different degrees on $F[x]$, $F$ a field. Let $f(x) \in F[x]$ f ...
4
votes
1answer
102 views

Is it true that in a Noetherian ring every descending chain of prime ideals stabilizes?

Is it true that in a Noetherian ring every descending chain of prime ideals stabilizes? It would be good if I had this result. As it would finish off my proof that the minimal primes of an ideal ...
1
vote
1answer
105 views

finding fixed points of frobenius endomorphism of a ring of char > $p$ (not a domain)

How can I find all the fixed points of the endomporphism $f \mapsto f^p$ in the ring $ \mathbb{F}_p/ (x^{p^n} - x)$. My hunch is that it should have something to do with $x^{p^n}-x$ being the ...
2
votes
3answers
54 views

Proof that a field is an integral domain

Here is my attempt at proving this. Let $F$ be a field and let $a \in F, \ a \neq 0$. Then $a$ is a unit and hence $\exists \ b \in F$ such that $ab = 1$ Now let $c \in F, \ c \neq 0$ Let $a \cdot ...
1
vote
3answers
79 views

Ring theory: Ideals being equal

Question: Prove directly, without gcd computations, the following equalities of ideals. (i) $(5, 7) = (1)$ in $\Bbb{Z}$ (of course (1) = $\Bbb Z$). (ii) $(15, 9) = (3)$ in $Z$. (iii) $(X^3 −1,X^2 ...
0
votes
0answers
58 views

Number rings and (round) parentheses versus (square) brackets [duplicate]

Is there a reason for the difference in the use of parentheses versus brackets as used in algebraic extensions. For example, when the field rational numbers ${\mathbb{Q}}$ extended with $i = ...
2
votes
1answer
66 views

A module with 300 elements

I have got this problem. Let it be $R=M_{2}(\mathbf{Z})$ the ring of square matrices over the integers. I need to find a $R$-module with $300$ elements and one question for this problem, can be there ...
4
votes
3answers
193 views

Irreducibility of a particular polynomial

I've got this problem for my homework: find out whether the polynomial $$f(x)=x(x-1)(x-2)(x-3)(x-4) - a$$ is irreducible over the rationals, where $a$ is integer which is congruent to $3$ modulo $5$. ...
1
vote
1answer
47 views

Fixed roots and quotient rings

I'm reading a paper by Smart and Vercauteren on homomorphic encryption (http://eprint.iacr.org/2011/133). I don't understand a specific statement around quotient rings. The authors state that for a ...
2
votes
2answers
63 views

On a ring $R$ where $x^2 = x$, we must have that $2x = 0$ and $R$ is commutative

I want to show that on a ring $R$ where $x^2 = x$ for all elements $x$, we must have that $2x = 0$ and $R$ is commutative. Does my answer look sound? Well, if $x = 0$ we have that $x^2 = x$ is ...
2
votes
1answer
45 views

Generator for the ideal $I + J$ where $I = (2 + 3i)$ and $J = (1 - i)$

On a related question I calculated the GCD of $I = (2 + 3i)$ and $J = (1 - i)$ to be $1$. Now I know that $\mathbb{Z}[i]$ is a principal ideal domain. And I also know that the greatest common divisor ...
1
vote
0answers
45 views

Every onesided nilideal of a right noetherian ring is nilpotent.

Suppose $R$ is a right noetherian ring. Prove that every onesided nilideal is nilpotent. I try to use this theorem: If R is a commutative Ring and I is nilideal of R and also I is finitely ...
2
votes
1answer
38 views

Equations over fields

Let $x_1,\cdots, x_n$ be distinct elements of a given field $F$ such that for any $k$, $\sum_{i=1}^n x_{i}^k = 0$. I want to show that all $x_i$'s are zero.
6
votes
2answers
64 views

Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$

Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$ Here is my attempt at solving this using a generalized Euclid's Algorithm. Does it look alright? Step 1 $2 + 3i = M(1-i) + N$ ...
2
votes
1answer
74 views

Definition of (co)homology of groups and Lie algebras: actions and augmentations

In the Chevalley-Eilenberg chain complex, what is $ux_i$? What does "trivial $\frak{g}$-module $k$" mean? Below I denote $R=k$ (any commutative unital ring). How is the augmentation (last map in the ...
1
vote
2answers
90 views

Prove that an idempotent $e$ of $R$ is primitive iff $\dim_{\,F}\left(\operatorname{Im} e\right)=1$

Let $V$ be a vector space over the field $F$, $R$ is the ring of linear operators on $V$. Prove that an idempotent $e$ of $R$ is primitive iff $\dim_{\,F}\left(\operatorname{Im} e\right)=1$ Thanks a ...
2
votes
1answer
144 views

Ring Elements are Either Nilpotent or Unital $\implies$ Ring has a Unique Prime Ideal

EDIT: I originally wrote the proof down wrong. Here is the corrected proof and question: Let $R$ be a commutative ring with identity. Suppose each element of $R$ is either a unit or a nilpotent ...
2
votes
1answer
128 views

Jacobson radical subset of maximal ideal

If we define the jacobson radical $J$ to be the intersection of all maximal right ideals then I am trying to show that if we have a maximal two sided ideal $M$ we must have $J\subseteq M$? Any ideas
2
votes
1answer
133 views

Every commutative subring is contained in a maximal commutative subring.

Let $R$ be a ring. Let $A$ be a commutative subring of $R$. Show that $A$ is contained in a maximal commutative subring of $R$. I'd like to use Zorn's Lemma. I'd like to start by proving that ...
0
votes
2answers
41 views

Isomorphism of rings under different operations

i am new to this site. i have been reading through its posts and question and they are really amazing. however, i found a link to a question asked about one year ago and a question i don't know how to ...
2
votes
3answers
235 views

References on Ring and Module Theory [duplicate]

Next semester I will take a course about Ring and Module Theory. Can anyone tell me the best texbooks and problems books about it. I only know some famous problems books such as Exercises in Classical ...
1
vote
1answer
70 views

Ideals generated by a set in a ring without multiplicative unity

It is well known that in a ring $R$ with $1\neq 0$, the ideal generated by $S$ is $$I(S)=\{a_1 s_1 b_1+a_2 s_2 b_2+\dots+a_n s_n b_n: n\in\mathbb{N}, a_i, b_i\in R, s_i\in S\}.$$ Is there a similar ...
2
votes
1answer
80 views

There exist $n\in\mathbb{N}$ such that $Imf^{n}+Kerf^{n}=M$

Let $M$ be a $R$-module, $f$ be $R$-automorphism on $M$. Prove that if $M$ satisfies DCC condition then there exist $n\in\mathbb{N}$ such that $Imf^{n}+Kerf^{n}=M$
1
vote
1answer
72 views

Identification of $\mathbb F_2[X]/(X^4+X+1)$

As mentioned in the title I would like to show that we can identify $\mathbb F_2[X]/(X^4+X+1)$ with the set $K$ of polynomials: $p_0+p_1 a+p_2a^2+p_3 a^3$ in a variable $a$ that we assume satisfies ...
2
votes
1answer
137 views

Is every regular element of a ring invertible?

When I am reading a paper, I found the definition of a new ring as following: In this definition, if every central regular element is invertible, i.e., how to understand the invertible element of u? ...
0
votes
1answer
87 views

Module over a ring satisfying ACC [duplicate]

Let $R$ be a ring that satisfies ACC on the set of its left ideals and $M$ be a finite generated $R$-module. Prove that every submodule of $M$ is finitely generated. I know that if $M$ ...
1
vote
1answer
84 views

group of automorphisms of the ring $\mathbb{Z}\times\mathbb{Z}$

Please help me with answering this question: Compute the group of automorphisms of the ring $\mathbb{Z}\times\mathbb{Z}$.
3
votes
1answer
56 views

Why does $\operatorname{Spec}(\prod_1^\infty \Bbb F_2)$ have connected components that are not open?

To be honest, I don't even know how to describe all prime ideals in $\prod_1^\infty \Bbb F_2$. I know we get one for each $n \in \Bbb N$ corresponding to the set of elements that are zero in the ...
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vote
2answers
700 views

Prove that if $R$ is a principal ideal ring and $S$ is a multiplicatively closed subset of $R$ then $S^{-1}R$ is also a principal ideal ring. [duplicate]

Prove that if $R$ is a principal ideal ring and $S$ is a multiplicatively closed subset of $R$ then $S^{-1}R$ is also a principal ideal ring. Thanks for any insight.
1
vote
1answer
75 views

Proof on ring isomorphism- irreducible

Consider the ring isomorphism $\phi: A \to B$ I have to prove that $a\in A $ is irreducible if and only if $\phi(a)$ is irreducible. By definition, $a$ is irreducible in A if and only if: 1) $a$ ...
0
votes
2answers
56 views

Why Must Prime Ring Elements Not be Units?

Let $p, a, b \in R$. We say that $p$ is a prime element of $R$ if (i) $p$ is not a unit and (ii) $p \mid ab \implies p \mid a$ or $p \mid b$. My question is why is it necessary to force $p$ to not ...
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votes
2answers
298 views

Let $I$ be a left ideal of a ring $R$. Prove that if $I$ is a direct summand then $I^2=I$.

Let $I$ be a left ideal of a ring $R$ with $1$. Prove that if $I$ is a direct summand then $I^2=I$. Definiton: An ideal $A$ is a direct summand of a ring $R$ iff there exist an ideal $B$ of $R$ ...
5
votes
2answers
207 views

$R$ has only one maximal ideal

Let $F$ be a field. Let $R$ be the set of all upper triangular matrices of the ring $M_{n}(F)$ with the property that its coefficients on the main diagonal are all the same. Prove that $R$ has only ...
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vote
2answers
36 views

$f(Nil_{*}R)\subseteq Nil_{*}S$ and there exist an epimorphism from $R/Nil_{*}R$ to $S/Nil_{*}S$

Let $f:R\rightarrow S$ be a ring epimorphism. Prove that $f(Nil_{*}R)\subseteq Nil_{*}S$ and there exist an epimorphism from $R/Nil_{*}R$ to $S/Nil_{*}S$ $Nil_{*}$ is prime radical. More precisely, ...
0
votes
1answer
34 views

$x\in R$ is a unit iff $(x+RadR)$ is a unit in $R/RadR$

Let $R$ be a commutative ring. Prove that the element $x\in R$ is a unit iff $(x+RadR)$ is a unit in $R/RadR$. ($RadR$ is Jacobson radical of $R$) Thanks in advanced.
1
vote
1answer
39 views

$R\cong\mathbb{Z}$ or $R\cong\mathbb{Z}/p\mathbb{Z}$ ($p$ prime)

Let $R$ be an integral domain and each subgroup of additive subgroup of $R$ forms an ideal of $R$. Prove that either $R\cong\mathbb{Z}$ or $R\cong \mathbb{Z}/p\mathbb{Z}$ ($p$ prime). Help me.
0
votes
1answer
261 views

Kernel of surjective ring homomorphism and induced isomorphism

Problem: Let $\phi: R\to S$ be a surjective ring homomorphism (the rings are not necessarily commutative) such that for every $a\in \ker\phi$ there is a $n\in\mathbb{N},n\ge 1$ s.t. $a^n=0$. ...
1
vote
3answers
40 views

$\mathbb{Z}_{p}\left[x\right]/\left\langle f\right\rangle $ is a semilocal ring.

Let $p$ be a prime, $f$ be a nonconstant polynomial that is contained in $\mathbb{Z}_{p}\left[x\right]$. Prove that $\mathbb{Z}_{p}\left[x\right]/\left\langle f\right\rangle $ is a semilocal ring.
2
votes
1answer
66 views

Prove that $Re+Rf=Re\oplus R(f-fe)$

Let $e,f$ be idempotent elements of a ring $R$. Prove that $Re+Rf=Re\oplus R(f-fe)$. This post solves the first part of my question. How should we prove the second part, that $e,f$ are ...