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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Boolean algebras and rings

I know that M. H. Stone proved that there is a bijection between boolean algebras and boolean rings. The bijection I know is the following: to any given Boolen algebra $(L,\, \vee, \wedge)$ we ...
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R has IBN but R fails rank condition

I need an example about "IBN for ring": R is a ring (no commutative), R has IBN but R fails rank condition. thanks
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ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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1answer
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Ring polynomial kernel generators

This is the textbook question: Q: Find generators for the kernels of the following maps: $\mathbb{R}[x,y] \to \mathbb{R}$ defined by $f(x,y) \mapsto f(0,0)$ $\mathbb{R}[x] \to \mathbb{C}$ defined ...
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1answer
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If $P$ is a prime then $R/P$ is an integral domain.

I know the same question has been already asked here. So, I am not asking for any proof rather to find out what's wrong with my proof. So, this is what I did: Let, $a+p, b+p \in R/P$, since $P$ is a ...
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1answer
29 views

Show that it is a homomorphism?

For any abelian group $G$ we have $e_n: G \to G, e_n(g) = g^n$. By convention $e_0(g) = 1$. For a Field $F$ we have the subgroup $\{1,-1\} \leq F^*$. When $F$ is of characteristic $2$, this is the ...
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2answers
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What Notation is this?

When $p$ is prime, show that $v: Z^*_p \rightarrow U_2$ I know that the $Z_p$ is the elements $\{0,1,2,\cdots,p-1\}$ But what about the star on top of the $p$? Is that the group operation? Because ...
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2answers
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Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel

I learned how to see quotient groups intuitively when I learned of a group mod its commutator subgroup. If we take a group and mod out all the elements that do not commute, we get a quotient group ...
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1answer
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Find the Number of Elements of a Particular Quotient Ring

Find the size of $\mathbb{Z}[\sqrt{-19}]/I$, where $I=(18+\sqrt{-19}, 7)$. The standard way to proceed would be $\mathbb{Z}[\sqrt{-19}]/I=\mathbb{Z}[x]/(x^2+19, 18+x, 7)=\mathbb{Z}_7[x]/(x^2+5, ...
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1answer
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How to find all roots of the equation $ x^3 + 2x^2 - 3x$ in $\mathbb Z_{12}$

Firstly you can factor it completely from $ x^3 + 2x^2 -3x$, which is $x(x-3)(x+1)$. We have the obvious roots of $0$, $3$ and $-1$, but what about the other roots? I have a little confusion here ...
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1answer
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Is $T_n(R) \cong T_n(R)^{op}$?

I am working on the following problem: Let $R$ be a commutative ring, and $T_n(R)$ be the ring of $n \times n$ upper triangular matrices. Is $T_n(R) \cong T_n(R)^{op}$? I have already shown ...
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Krull's height theorem in the non-Noetherian case

Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $\operatorname{ht} I\le n$. When $R$ is not Noetherian, this is not ...
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2answers
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Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
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2answers
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Simple generator modules

Let a ring $R$ with identity element be such that the category of left $R$-modules has a simple generator $T$. My question: "Is $T$ isomorphic with any simple left $R$-module $M$?" I tried the ...
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Question about the Chinese Remainder Theorem and the residue class ring ${\bf Z}/p\# {\bf Z}$

In a question that I asked on MO, Terence Tao observed that: The Chinese Remainder Theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of ...
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Isomorphism and Quotient Ring [on hold]

Let $R$ be a ring. If for every proper ideal $I$ of $R$ we have $R/I\cong R$, then show that for every two proper ideals $I$ and $J$ of $R$ either $I\subseteq J$ or $J\subseteq I$.
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1answer
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Why do we have a basis?

A corollary that is in my book that I think is relevant to my question is: If E is an extension field of F, $\alpha \in E$ is algebraic over F, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ ...
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Need Help Understanding Why Proof Shows Set is not a Ring

I am having trouble reading this somewhat "slick" proof. Maybe it's not as slick as I think it is though, and I'm missing something here. So, I understand everything that is being done until the last ...
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$x\in\mathcal{O}_K$ can be written as product of irreducible elements

Lemma: Every element $x\neq 0$ of $\mathcal{O}_K$ with ($K$ an arbitrary number field) can be written as a product of irreducible elements. Proof: We prove this lemma by complete induction on ...
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1answer
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What is the difference between these two conditions $J = \{az \mid a \in R\}$ and $ I = \{a \in R \mid az \in J\}$

Please consider these two questions: Let $R$ be a ring and $z \in R$, which is fixed. Let, $J = \{az \mid a \in R\}$. Prove that $J$ is a left ideal of $R$. Skipping the subtraction part, this is ...
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3answers
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Why is it the smallest subfield containing F and $\alpha$?

Please take a look at the sentence in red: I understand that $\phi_\alpha[F[x]]$, is a subfield which contains $\alpha$, and F(we just need to evaluate $\phi_\alpha$ at the appropriate values). But ...
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1answer
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The annihilator induces a module [duplicate]

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
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Question about the deduction of the quotient ring $R/I$

Yesterday we deduced on class how quotient groups were deduced and well defined. Let $R$ be a ring and $I$ an ideal of $R$. My professor proved us that the multiplication operation $$R/I \times R/I ...
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3answers
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Prove or disprove $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$

Want to prove or disprove this $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$ as communtative rings. I can show that $x^5-3$ and $x^5-9$ are irreducible in $\mathbb{Q}$, but I cannot go from ...
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Show that if $R$ and $S$ are ideal of a ring $A$ then the product $R\cdot S$ is a ideal of $A$. [duplicate]

How to prove that if $R$ and $S$ are ideal of a ring $A$ then the product $R\cdot S$ is a ideal. I can't show only that if $x, y\in R\cdot S$ then $x-y\in R\cdot S$. The other axioms of ideal I ...
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3answers
35 views

Example of ideal generated by two elements

I have an easy example on my notes that I don't understand. My teacher said that in $\mathbb{Z}$, $(2,3)=2\mathbb{Z}+3\mathbb{Z}$ is a principal ideal, because $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$. ...
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1answer
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Is group of units of a polynomial ring only constant polynomial which is involved in R

Let R be a integral domain(or maybe field) edit : Let R be a field. The group of units of R[x] is $$ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots+ a_2 x^2 + a_1 x + a_0 $$(or infinity) such ...
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Commutative ring of prime power order

Suppose $R$ is a non trivial commutative ring with identity of prime power order. What can we say about the structure of $R$? If $R$ is of prime order, then $R$ is a field?
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$x^{mn} -a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible.

Let F be any field, a is in F and (m,n)=1. Show that $x^{mn}-a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible in F[x]?
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if the sum of two units is a unit, then there is an unique maximal ideal

Let $R$ be a ring with identity element. I have to proof that if the sum of two units of $R$ is a unit, then $R$ has an unique maximal ideal. But i don't see a connection. If someone could give me a ...
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1answer
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For a field $K$ is $K\subset{K[X_{1},…X_{n}]}$

Let $K$ be any field and $K[X_1,...X_n]$ the ring of polynomials in $X_1,...X_n$ with coefficients in $K$. I am wondering if $K$ is a subset of $K[X_1,...X_n]$. I believe $K\subset{K[X_1]}$ since ...
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1answer
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Prove that $R$ has ACCP if and only if every non-empty collection of principal ideals of $R$ has a maximal element.

Let $R$ be a commutative ring. Prove that $R$ has ACCP if and only if every non-empty collection of principal ideals of $R$ has a maximal element. Prove further that if $R$ is an integral domain and ...
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1answer
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The norm of Gaussian integers and the irreducible element $ 1 + i $.

Note: Let $ \text{N}(a + bi) \stackrel{\text{df}}{=} a^{2} + b^{2} $. Observe that $ \text{N}(1 + i) = 2 $. Is it always true that if $ 1 + i $ divides a Gaussian integer, then the norm of $ 1 + i $ ...
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1answer
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Quotient of the ring of integers of a quadratic field by the ideal generated by a split integer prime.

I am wondering about primes $p$ in $\mathbb Z$ that are split in $\mathcal O_{K}$, $K=\mathbb Q(\sqrt d)$. Let $\omega=\sqrt d$ if $d \equiv 2,3 \mod 4$ and $\omega=\frac{1+\sqrt d}{2}$ if $d \equiv 1 ...
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What are all the integral domains that are not division rings?

A commutative division ring is an integral domain. But what are all the integral domains that are not division rings? The examples I currently know are the following: $\mathbb{Z}$, $\mathbb{Z}[i]$, ...
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1answer
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How to prove subfield generated $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n) $

This is problem in Hungerford chapter 5: Fields and Galois Theory. Prove $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n)$ and $K[u_1,u_2,..u_{n-1},u_n]=K[u_1,u_2,..u_{n-1}][u_n] $ My ...
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1answer
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Show that $m = \pm 2$ or $m = \pm 3.$

Let $$R = \left\{\frac{a + b\sqrt{-19}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\} = \mathbb{Z} \left[\dfrac{1+\sqrt{-19}}{2} \right] = \mathbb{Z}[\alpha].$$ and define $d:R \setminus \{0\} ...
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Let $q$ be a prime congruent to 3 mod 4, prove the quotient ring $\mathbb{Z}[i]/(q)$ is a field with $q^2$ elements

Let $q$ be a prime congruent to 3 mod 4, prove the quotient ring $\mathbb{Z}[i]/(q)$ is a field with $q^2$ elements The field portion I understand. $\mathbb{Z}[i]$ is a PID and because $q$ is ...
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Let $\mathbb{Z}[i]$ denote the Gaussian integers. The set of units of $\mathbb{Z}[i]$ is $\{\pm 1, \pm i\}.$

Let $\mathbb{Z}[i]$ denote the Gaussian integers. The set of units of $\mathbb{Z}[i]$ is $\{\pm 1, \pm i\}.$ A proof from: https://proofwiki.org/wiki/Units_of_Gaussian_Integers (Proof 2) Proof. ...
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1answer
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Prove that if $z$ is good then so is $z + r$ for every $r \in R$.

Let $$R = \left\{\frac{a + b\sqrt{-19}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\} = \mathbb{Z} \left[\dfrac{1+\sqrt{-19}}{2} \right] = \mathbb{Z}[\alpha].$$ Note that $R$ is an integral ...
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1answer
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Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$ [duplicate]

Let $R$ be a ring. An element $r \in R$ is called nilpotent if $r^n=0$ for some integer $n \ge 1$. Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$. Proof. Recall ...
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What is $Q(x)$?

I do not really understand what $\mathbb{Q}(\pi)$ is here: Ofcourse we see that $\mathbb{Q}(\pi)$ is a field. But I have to "guesses" of what they mean, is one of them correct? 1. ...
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Let $R$ be a ring with identity. Prove that if $1-ab$ is invertible for some $a,b \in R$, then $1-ba$ is also invertible. [duplicate]

Let $R$ be a ring with identity. Prove that if $1-ab$ is invertible for some $a,b \in R$, then $1-ba$ is also invertible. Ok, si if $R$ is a ring with unity, then we have $R$ with $1 \ne 0$ We have ...
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Direct product of Rings isomorphism

I was reading chapter five in my Abstrat algebra book about finite Abelian groups. In Proposition 6 part (1). It states that $Z_{m} \times Z_{n} \cong Z_{mn}$ if and only if $gcd(m,n)=1$. This ...
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Is $f(x)$ reducible if $f(a)=0$

I am confused about this seemingly trivial question: If $f(a) = 0$ for some $a\in D$, then when is $f(x)$ reducible in $D[x]$? ($D$ is an integral domain). My answer: Always. Let $f(a)=0$. ...
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Find elements in the center of $n × n$ matrix ring $M_n (R) $ for any $n ≥ 2$. [duplicate]

Let $R$ be a ring. The center of $R$ is the set $C(R) = \{c ∈ R : cr = rc, ∀r ∈ R\}$. Determine elements in the center of the $n × n$ matrix ring $M_n (R) $ for any $n ≥ 2$. So, we have that ...
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Uniqueness of prime ideals of $\mathbb F_p[x]/(x^2)$

What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ ...
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Group Ring over commutative ring

$R$ is commutative semiprime ring, $(R,+)$ abelian group without torsion. Then $RG$ is semiprime. Proof by contradiction. If $x$ is nilpotent element in $RG$, then $x=r_1g_1+r_2g_2+...r_ng_n$. ...
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1answer
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Factoring the Ring of Integers into Ideals

Let $K$ be a number field. Let $\frak p$ be a prime ideal in $\mathcal O_K$. Let $u\in \mathcal O_K$ and $m\in \mathbb N$. I've been told that $|u|_{\frak p} = |m|_{\frak p} = 1$ where $|\cdot|_{\frak ...
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1answer
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+500

Fermat's last theorem and $\mathbb{Z}[\xi]$

I heard that one can prove special cases of FLT by using unique factorization in $\mathbb{Z}[\xi]$ (whenever this is possible), where $\xi$ is a primitive $n$-th root of unity. How can one do this in ...