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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Generalized fact in ring theory about irreducible elements

It is quite easy to show that for $A$ an integral domain, an element $a \in A$ is irreducible if and only if the principal ideal $\langle a \rangle$ is maximal for inclusion among proper principal ...
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Idempotent semiring

Let $R$ be a semiring. For $a\in R$,we define $t_a(x)=x+a$ for all $x\in R$. Prove that $R$ is idempotent(with +) and $1$ has an infinite order if and only if for all $a,x,y\in R$, ...
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1answer
54 views

The existence of a polynomial factor

Given two polynomials $p_1(x_1,\dots, x_m)$ and $p_2(x_1,\dots, x_n)$ over reals, where $m > n$, and we know that $p_2(x_1,\dots, x_n)=0 \implies p_1(x_1,\dots, x_m) =0$. My question is: ...
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1answer
32 views

Let $\Phi : R \rightarrow R'$ be a ring homomorphism, where $R,R'$ are rings with unity. Then which of these is true?

Let $\Phi : R \rightarrow R'$ be a ring homomorphism, where $R,R'$ are rings with unity. Then which of these is true : $(i)~\Phi(1)=1 ~\forall~$ rings $R,R~'$ with unity $(ii)~\Phi(1) \ne 1 $ for ...
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1answer
49 views

Principal ideal domain, $\forall x=(x_1,x_2)^t \in R^2~~\exists G \in SL_2(R) : Gx=(\gcd(x_1,x_2), 0)^t$

Let $R$ be a principal ideal domain. Prove that for every $x=(x_1,x_2)^t \in R^2$ exists a matrix $G \in SL_2(R)$ for which $Gx=(\gcd(x_1,x_2), 0)^t$. I think it's easy, but do not know how to ...
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2answers
83 views

Can We Always Build a Field out of an Integral Domain?

Link to Hungerford's Text Let $R$ be an integral domain, and $F$ its quotient field (or field of fractions). Assuming that $\phi: R \rightarrow F$ is isomorphic, $R[x]$ is isomorphic with $F[x]$ ...
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1answer
62 views

Does the ring of continuous functions determine $\mathbb R^n$?

I have two related questions which are just making the question asked in the title more specific: (a) Is every ring homomorphism (or maybe $\mathbb R$-algebra homorphism) between rings of the form ...
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2answers
176 views

Is the zero ring a domain?

Is the zero ring usually considered a domain or not? Wikipedia says: The zero ring is not an integral domain; this agrees with the fact that its zero ideal is not prime. Whether the zero ring ...
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1answer
21 views

Algebra. If $R$ is a commutative associative ring with neutral element.

Algebra. If $R$ is a commutative associative ring with neutral element. Then if R|P is an integral domain ($ab \in R|P \implies a=0 \ or \ b=0$) , then $1 \notin $ P ? This is the only thing I can ...
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3answers
170 views

If $x ^ 6 = x$, prove that $x ^ 2 = x$, in a ring

I found a short and interesting problem: given a ring $(R, +, ⋅)$ and knowing that $x ^ 6 = x\ (\forall x\in R)$, prove that $x ^ 2 = x$ (∀ x ∈ R). While it is short, I cannot figure out how to solve ...
9
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1answer
169 views

Ring of real-valued convergent sequences

Here is a fun and challenging problem: Let $R$ denote the ring of real-valued convergent sequences and let $S$ denote the ring of real-valued sequences. Prove or disprove that $S\cong R$. ...
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1answer
44 views

Looking for group of polynomials with only real roots

Assume $P_\mathbb R$ is the set of all polynomials which have only real coefficients and only real roots. Define $0$ as a polynomial with infinitely many real roots and all other constant polynomials ...
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1answer
20 views

Prove Z_m*n has at least 2 idempotent elements besides 0 and 1. [duplicate]

Here m, n are relatively prime and greater than 1. Z_mn is the ring of nonnegative integers less than mn under modulo m*n addition and multiplication. An idempotent element a is a ring element with ...
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3answers
197 views

Does an ideal of a ring contain the zero element of the additive group of the ring? If yes does this apply to the multiplicative group?

Well, i was wondering about this, because this is the only thing i can deduct right now in a proof in reading. If it turns out the this is not the case i will post the detail in the proof that is ...
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1answer
37 views

Suppose $\phi:K[x]\to K[x]$ is an automorphism such that $\phi(u)=u\ \forall u\in K$. Prove that $\phi(x)=ax+b$.

Let $K$ be a field. Suppose $\phi:K[x]\to K[x]$ is an automorphism such that $\phi(u)=u\ \forall u\in K$. Prove that $\phi(x)=ax+b$. Since $K$ is a field then $K[x]$ is an Euclidean Domain.What ...
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0answers
7 views

Ring of convergent power series is discrete valuation ring

Suppose $\Theta_U$ is the ring of homomorphism and U always contains 0. I need to show that $\varinjlim \Theta _U$ is a discrete valuation ring, so I need to find discrete valuation. If the least ...
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1answer
45 views

When $f(I)S=S$ for each ideal $I$ of $R$?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). Question 1. ...
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1answer
25 views

Prime ideals in a Dedekind domain

If $R$ is a Dedekind domain and $I\subset R$ is a non-zero ideal then by the Noetherian property of $R$, I can show that there are distinct non-zero prime ideals $P_1,...,P_r$ s.t. $P_1^{a_1}\cdots ...
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1answer
45 views

Show that the $Y^3-Y+1$ splits completely into linear factors in $E[Y]$ and find these factors

I am working through some past exam papers, and I have some conceptual questions that I need to clear up regarding this question: The full question is: Let $\mathbb{F_3}= ...
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0answers
26 views

Multiplicative Inverse of Polynomials in Finite field

Find the multiplicative inverse of $x + 2$ in the field $\Bbb Z_5[x]/(x^2 + 2)$. I have done the following so far: \begin{align*} x^2+2 &= (x+2)(x+3) + 1\\ (x+2)(x+3) &\equiv -1 \pmod ...
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1answer
24 views

Localization of ideals at all primes

Let $R$ be a commutative ring with $1$ and $I$, $J$ ideals in $R$. For a prime ideal $P$, let $I_P=(R-P)^{-1}I$ be the localization of $I$ at $P$. Question: If $I_P=J_P$ for all prime ideals ...
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1answer
25 views

The invertible elements of $\mathbb{Z}(\sqrt{-2})$

Let $\mathbb{Z}(\sqrt{-2}):=\{a+b\sqrt{2}i; \, a,b \in \mathbb{Z} \}$. Find the multiplicative inverses. My attempt: We write $(a+b \sqrt{2}i)(c+d \sqrt{2}i)=1$ It trivially follows: ...
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1answer
25 views

Check the irreducibility of the polynomial over $\mathbb Z$

$f_1(x) = x^5 + 5x^2 + 1$ $f_2(x) = x^4 - x^3 + 14 x^2 + 5x + 16$ Use the result if $p(x)$ is a monic polynomial with integer coefficient and $p(d) \neq 0$ for all integer $d$ dividing the constant ...
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0answers
53 views

homology of mapping telescope of a monoid

Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ ...
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0answers
45 views

Express the ideal $(6) \subset\mathbb Z[\sqrt {-5}]$ as a product of prime ideals.

Express the ideal $(6) \subset\mathbb Z[\sqrt {-5}]$ as a product of prime ideals. I know I can write $(6)=(2)(3)=(1+\sqrt {-5})(1-\sqrt {-5})$. But I guess these factors might not be prime. ...
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3answers
79 views

If the intersection of ideals $I_{1},\ldots,I_{n}$ is contained in a prime ideal $P$, then one of them is contained in $P$

Let $A$ be a commutative ring and $I_{1},\ldots, I_{n}$ and $P$ ideals in $A$ with $P$ prime so that $\cap_{i=1} ^{n} I_{i} \subset P $. Show that there's an $i_0 \in \{1,...,n \}$ so that $I_{i_0} ...
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1answer
57 views

Binomial rings closed under colimits?

A binomial ring is a ring (for the purposes of this question all rings are commutative and unital) which is torsion-free and has, for each $n$, a binomial function $\binom{x}{n}$ satisfying ...
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2answers
36 views

Torsion of $\mathbb Z ^{2 \times 2}$

I am trying to find the torsion of the ring $R= \mathbb Z ^{2 \times 2}$ seen as an $R-$ left module. So I want to see for which matrices $B$, there exists $A \neq 0$ such that $AB=0$. So if ...
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1answer
38 views

Endomorphism ring of $x \Bbbk\langle x,y \rangle + y \Bbbk\langle x,y \rangle$

Let $R = \Bbbk \langle x,y \rangle$, where $\Bbbk$ is a field. I want to determine $\underline{\text{End}}_R(xR + yR)$, the ring of (not necessarily degree-preserving) graded module homomorphisms ...
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1answer
51 views

Characteristic of nonzero commutative rings with unity

Let $\mathcal R$ and $\mathcal S$ be non zero commutative rings with unity. Then: char($\mathcal R$) is always a prime number $\mathbb{Q,R}$ contradicts this. ($char(\mathcal R)=0$) ...
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1answer
35 views

Prove S is a subring of R.

Let a be an element of ring R. Let S = {x in R:ax=0}. Prove S is a subring of R. First we will show that S is a subgroup of R under the addition of R. We let x,y be arbitrary elements in R. Then ...
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1answer
54 views

Units in a Euclidean domain

I want to prove that if $R$ is a Euclidean domain, with Euclidean function $d$, then if $a,b\in R\setminus\{0\}$ and $b$ is not a unit then $d(a)<d(ab)$. I already proved that $b$ is not a ...
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3answers
77 views

Prove that if $b \mid c$ then $ab \mid c$?

This is an exercise in a text I am reading. Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose that $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. ...
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1answer
42 views

Generators of $\mathbb{Z}[X_1,\ldots,X_m]$

I was wondering about the following (possibly stupid) question: if $S$ is a one element (ring) generator set of $\mathbb{Z}[X]$, $S$ must consist of a linear polynomial in $X$ (= degree $1$). Is the ...
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2answers
68 views

Number of elements in the quotient ring $\mathbb{Z}_6 [x]/\langle 2x +4\rangle$ [duplicate]

I am confused about the quotient ring. I know that $\mathbb{Z}_7 [x]/\langle x^2 + 1\rangle = \{ f(x) + \langle x^2 + 1\rangle \mid f(x) \in \mathbb{Z}_7[x] \}$. Here $x^2 + 1$ is a zero element in ...
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1answer
49 views

Inverses of elements in group algebras

If $G$ is a finite group whose elements are $g_1,\ldots,g_n$ and let $F$ be the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. We define a vector space over $F$ with ...
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4answers
211 views

Rings of p adic integers contains copy of the integers

I need to prove that p adic integers contains copy of integers. I proved this way: For integer $n=n_0$, we can define $n_i$ as $n_i = n_{i+1} p^{i+1}+ a_i$ where $a_i$ is between $0 $ and $p-1$. ...
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2answers
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Does $I(J\cap K)=IJ\cap IK$ hold in a finitely generated polynomial $K$-algebra for $K$ a field?

Let $K$ be a field and $R:=K[X_1,X_2,\cdots, X_n]$ for a certain $n\in\mathbb N$. If $I,J,K$ are three ideals of $R$, can we conclude that $I(J\cap K)=IJ\cap IK$?
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1answer
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If $R\subset S$, does it necessarily follow that $R[G]\subset S[G]$?

So perhaps this is a trivial question, but suppose we have some rings (possibly noncommutative) such that $R\subset S$, and a group (possibly infinite) $G$, does it necessarily follow that ...
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4answers
104 views

$\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$

I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to $\mathbb Z_5$. The only thing that came to my mind was trying to apply the first isomorphism theorem using an ...
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0answers
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Showing function defined on $\text{Frac}(R)$ is a ring homomorphism

Let $f : R \to S$ be a ring homomorphism where $R, S$ are integral domains. I want to show that $\varphi : \text{Frac}(R) \to \text{Frac}(S)$ defined by $r/1 \mapsto f(r)/1$ is a ring homomorphism. ...
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1answer
45 views

How do I find the quotient field of $\mathbb{Z}[\sqrt{d}]$?

Our teacher said sometimes the quotient field is $\mathbb{Q}[\sqrt{d}]$ and sometimes it's $\mathbb{Q}[\frac{1+\sqrt{d}}{2}]$. How do we decide, or what are the conditions on $d$ which helps us to ...
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1answer
82 views

Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?

Fix a positive integer $p$, possibly prime. For each natural number $n$, there is a ring $\mathbb{Z}/p^n \mathbb{Z}$ together with a distinguished ring homomorphism $$\pi_n:\mathbb{Z} \rightarrow ...
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1answer
32 views

What are irreducibles in the ring $R$ of entire functions?

Here's how far I have gone: I first find the units in $R$. Let $f$ $ \in $ $R$ be a unit. Then $\exists$ $g$ $\in$ $R$ such that $fg=1$, i.e $f(z)g(z)=1$ $\forall$ $z$ $\in$ $\Bbb C$. Then, $f(z)$ ...
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1answer
49 views

finding all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$, faster method?

I want to find all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$ and I wonder if there's a better method than the following. The polynomials are of form $aX^2 + bX +c$. So I have to ...
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2answers
31 views

generating element of $I = \{p \in \mathbb{Q}[X]: p(0)=0, p'(0)=0\}$

I have the ideal $I = \{p \in \mathbb{Q}[X]: p(0)=0, p'(0)=0\}$. I have verified that it is an ideal by multipyling an arbitrary element of the ideal with an arbitrary element of $\mathbb{Q}[X]$ and ...
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1answer
33 views

Exactly one ring homomorphism $F[X] \rightarrow S$

Let $F$ be a field, and $f \in F[X]/(f)$. Let $f$ have a zero point $\alpha$, that is, $f(\alpha)=0$. Let $F$ be a subring of $S$, and $\beta \in S$ with $f(\beta)=0$. Show that there is exactly one ...
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1answer
84 views

What is the kernel of $R[T] \to R[w]$, $T \mapsto w$, $w=u/v$, $u,v \in R$, where $R$ is an integrally closed domain?

I am posting the following question after posting a similar question: What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$? If $R$ is an integral domain, $w=u/v$, where $u,v ...
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3answers
40 views

Prime ideal in $\mathbb Z[\sqrt{10}]$

I am trying to solve this exercise: Prove that $\langle 2,\sqrt{10} \rangle$ is a prime ideal in $\mathbb Z[\sqrt{10}]$. I could do the following: I pick an element of the form $zw \in \langle ...
5
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1answer
86 views

What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$. Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...