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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Prove that $a$ is a prime element of $R$

Let $R$ be a PID and $P = (a)$ is a prime ideal of $R$. Prove that $a$ is a prime element of $R$. Since $P$ is a prime ideal of $R$, let $x,y \in R$ s.t. $xy \in P = (a).$ (WTS $a \mid x$ or $a\mid ...
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Exercise of these topics!

I am looking for more exercise to practice on these topic rings modular arithmetic isomorphism homomorphism
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Ring Homomorphism Textbook Question

Please help me understand the last three sentences in this paragraph from the Artin textbook. Where does this come from: "The monomials that appear in $r_0(t^2)$ have even degree, while those in ...
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Ring Homomorphism Defintion

$\varphi:R\rightarrow S$ is said to be a ring homomorphism if, $R,S$ are rings and $\varphi$ is a map such that: $\varphi(r_{1}+r_{2})= \varphi(r_{1})+\varphi(r_{2})$, $\varphi(r_{1}.r_{2})= ...
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A given ring of matrices has an infinite number of invertible elements

The set $\mathcal{M} = \bigg\{ \begin{pmatrix} a & 2b \\ b & a \\ \end{pmatrix} \bigg\vert a,b \in \mathbb{Z} \bigg\}$ is given. Prove that: (1) $\mathcal{M}$ is a commutative ring with ...
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1answer
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Least common multiple for integer matrices

Given two full-rank $3\times3$ integer matrices $M_1$ and $M_2$, I am trying to find integer matrices $N_1$ and $N_2$ such that $M_1N_1$=$M_2N_2$, such that $\left|\det(M_1N_1)\right|$ is minimal. ...
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1answer
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How to define the isomorphism?

Let $R$ be a ring, then For $R[x]/\langle x-1\rangle \cong R$, we define the map, $\varphi$ : $R[x]\rightarrow R$, defined by $\varphi(f) =f(1)$ For $R[x]/\langle x\rangle \cong R$, we define the ...
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1answer
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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 [on hold]

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
15 views

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
33 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
59 views

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
56 views

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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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
21 views

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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33 views

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
29 views

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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30 views

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
26 views

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
37 views

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 ...
3
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1answer
35 views

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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0answers
31 views

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 ...