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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1answer
24 views

Proving that the intersection of two subrings of R is also a subring of R

If $R_1$ and $R_2$ are both subrings of $R$ , how to prove that $R_1 \cap R_2$ is also a subring of $R$. here is my attempt (1) since $R_1$ is a subring of $R$ then it must contain zero (identity ...
4
votes
2answers
73 views

Factoring in $\mathbb{Z}[\sqrt{2}]$

How would one factor a number, say $9+4\sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$? This is what I've attemped to do: $$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2}) $$ $$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$ ...
0
votes
1answer
29 views

Is there an example that a unit divisible by a prime element?

Let $R$ be a UFD. Let $u$ be a unit in $R$ and $p$ be a prime element of $R$. Is it possible that $p|u$?
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0answers
38 views

List or characterize all of the units in each of the following rings

For $Z_{10}=\{0,1,2,3,4,5,6,7,8,9\}$ And the definition of a unit a is such that $a\times a^{-1} = 1 $ The answer seems to be $\{1,3,7,9\}$ but $2\times4 = 8 \bmod{7}$ congruent to $1$ so why isn't ...
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2answers
72 views

is $f(x)=x^4 +x^3 +x^2 +x +1$ irreducible in $R=(\mathbb Z/7\mathbb Z)[x]$?

is $f(x)=x^4 +x^3 +x^2 +x +1$ irreducible in $R=(\mathbb Z/7\mathbb Z)[x]$? I know that $f(x)$ cannot be expressed as a product of degree 1 polynomial and degree 3 polynomial since it has no roots in ...
1
vote
1answer
67 views

Length of a module over a local Artinian ring

If $R$ is a local Artinian ring with maximal ideal $P$, and $M$ is a finitely generated $R$-module, I would like to show that the length of any series $M= M_n\geq M_{n-1}\geq\dots\geq M_0=(0)$ such ...
0
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1answer
26 views

Is a ring $R$ factorial $\iff$ $R[X]$ factorial?

Let $R$ be a factorial ring. Then, the polynomial ring $R[X]$ is factorial. I was wondering if the other direction also works (i.e. $R[X]$ factorial $\implies$ $R$ factorial)? If not, please give ...
0
votes
2answers
29 views

Property of unfaithful module over PID

Proposition Let $R$ be a PID and $M$ a torsion module over $R$. Suppose $M$ is not faithful, with $\operatorname{Ann(M)}=Ra$ and $a=u{p_1}^{\alpha_1}...{p_n}^{\alpha_n}$ an irreducible factorization ...
0
votes
2answers
66 views

Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$

Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$. So If I understand correctly, I start with $x^2=12$ and then you ...
2
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1answer
25 views

Trace ideal of generator

Let $R$ be a unital ring. In Lam's "Lectures on modules on rings" (Theorem 18.8, p483), the following implication is stated for a right $R$-module $P$: $$\mathrm{tr}(P) = R \implies R \text{ is a ...
1
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1answer
18 views

Uniqueness of inverses in rings.

I can't get example of ring where inverses aren't unique ? What is condition required for the inverses to be unique . Kindly help..
3
votes
2answers
93 views

PID and finitely generated module

I am trying to prove the following statements: Let $R$ be a PID and $M$ a finitely generated $R$-module. Prove: (a) $M$ is torsion module iff $\operatorname{Hom}_R(M,R)=0$ (b) $M$ is an ...
2
votes
3answers
137 views

How can I find the kernel of $\phi$?

We have the homomorhism $\phi: \mathbb{C}[x,y] \to \mathbb{C}$ with $\phi(z)=z, \forall z \in \mathbb{C}, \phi(x)=1, \phi(y)=0$. I have shown that for $p(x,y)=a_0+\sum_{k,\lambda=1}^m a_{k \lambda} ...
0
votes
1answer
49 views

Determining whether a set is a ring?

Our homework problem asked us to determine whether the following set is a ring or not. My friend is telling me that it's not a ring, but I'm a bit confused about which condition it doesn't satisfy? ...
1
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2answers
183 views

Ideals-algebraic set

Notice that in $\mathbb{C}[X,Y,Z]$: $$V(Y-X^2,Z-X^3) = \{ (t,t^2,t^3) \mid t \in \mathbb{C}\}$$ In addition, show that: $$I(V(Y-X^2,Z-X^3)) = \langle Y-X^2,Z-X^3 \rangle$$ Finally, prove that the ...
1
vote
1answer
36 views

Quotientring-Show the isomorphism

To show that $\mathbb{C}[x,y]/\langle x,y \rangle \cong \mathbb{C}$ I did: $p(x,y) \in \mathbb{C}[x,y]$ $p(x,y)=\sum_{m,n=0}^k a_{mn}x^m y^n$ modulo $\langle x, y \rangle$ we have that $x \equiv ...
3
votes
2answers
63 views

Question about the muliplicative identity, inverse elements and nonzero elements of a ring?

Let R be the set of continuous real-valued functions on the interval [0, 1]. Show that R is a ring with respect to the operations (f + g)(x) = f(x) + g(x) and (fg)(x) = f(x)g(x). So my homework ...
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1answer
41 views

Is there anyone who could help me with this problems? [closed]

Let $R=\mathbb{Z}[\sqrt7]=\{ a+b\sqrt7 ~| a,b\in \mathbb{Z}\} $ define addition and multiplication by $(a_1+b_1\sqrt7)+(a_2+b_2\sqrt7)=(a_1+b_2)+(b_1+b_2)\sqrt7$ ...
0
votes
1answer
71 views

Should it stand that $\gcd(f(x), g(x))=1$?

If we have an ideal of the form $I=\langle f(x), g(x)\rangle\subseteq\Bbb Z[x] $ should it stand that $I=\langle \gcd(f(x),g(x))\rangle$? For example, if we have the ideal $I=\langle 2,x \rangle $ ...
0
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1answer
28 views

Some properties of matrices over rings

Problem (i) Let $R,T$ be division rings and $m,n \in \mathbb N$, then $M_n(R \times T) \cong M_n(R) \times M_n(T)$ and $M_m(M_n(R)) \cong M_{mn}(R)$. (ii) If $R$ is a semisimple ring and $n \in ...
2
votes
3answers
23 views

Torsion-free module morphism

I am trying to prove the statement: Let $R$ be a PID but not a field and let $M$ be an $R$-module. Then $$ M \space \text{is torsion-free $R$-module} \space \text{iff} \space ...
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votes
2answers
45 views

Confusion with definition of irreducible.

An integer $p$ is said to be irreducible if whenever $p=ab$ then $a $ or $b$ is $1$ or $-1$. Then we define an irreducible element $p$ in a commutative ring $R$ with unity as: $1)$ $p \neq 0$ ...
1
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1answer
40 views

If A , B are finitely generated R-algebras then $A\otimes_RB$ is a finitely generated $R$-algebra.

$A$, $B$ are finitely generated $R$-algebras. $R$ is a commutative ring with $1$. Then how can I show that $A\otimes_RB$ is finitely generated $R$-algebra? What I have tried: First I have to show ...
2
votes
1answer
53 views

PID and module problem

Let $R$ be a principal ideal domain but not a field, and let $M$ be an $R$-module. Show the following: (i) Let $p \in R$ be an irreducible element and $r \in R \setminus \{0\}$. Then $(R/ ...
4
votes
5answers
119 views

Determine if $\mathbb R[X,Y]/I$ is a field

Let $I=\langle Y+X^2-1,XY-2Y^2+2Y \rangle$, decide whether if $\mathbb R[X,Y]/I$ is a field. What I've tried to do was the following: If $\mathbb R[X,Y]/I$ is a field, then $I$ is a maximal which ...
1
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1answer
46 views

Lemma about a prime times a unit [duplicate]

I came across this Lemma: "Let $R$ be an integral domain, and let $a,u\in R$ such that $u$ is invertible. Then $a$ is a prime if and only if $au$ is a prime. I tried to prove it unsuccessfully, but ...
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2answers
32 views

show the following homomorphism is an isomorphism

pretty sure one-to-one correspondence means bijective. This question is from Undergraduate Commutative Algebra by Miles Reid. I have a feeling that this is wrong, but maybe my definition of ...
0
votes
1answer
39 views

Simultaneous congruences

Let $\mathbb K$ be a finite field and $\mathbb K[x, y]$ the polynomial ring in the commuting indeterminates $x$ and $y$. Consider the factor ring $\mathbb K[x, y]/\langle x^3, y^3\rangle $. Can we ...
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1answer
57 views

Why $\mathbb{Z}[\sqrt{-3}]$ is not a Euclidean domain? [duplicate]

I need to prove that $\mathbb{Z}[\sqrt{-3}]$ is not a Euclidean domain. I tried to show that $\mathbb{Z}[\sqrt{-3}]$ is not a P.I.D. but all ideals that I generate by two elements, turn out to be ...
1
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2answers
31 views

how can I prove that $D^{-1}R$ has a unique maximal ideal?

Let $R$ be a commutative ring without zero divisors & let $P$ be a prime ideal. I have shown that $D=R\setminus P$ is a non-empty multiplicative closed set without zero-divisors. Now how can I ...
3
votes
2answers
47 views

Find element in factor ring

The problem is this: 1) Prove that $f = x^3+x+1$ is irreducible over $\mathbb Z_5$. 2) In field K = $\mathbb Z_5/(f)$ $c = x+(f)$. Find in $K$ element $(c^2 + c + 1)^{-1}$ and write it in the form ...
1
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1answer
71 views

Is this “sliding window” unique?

Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists. Set $p$ a prime greater than 2, $\alpha = \lfloor p/2 \rfloor$, ...
3
votes
3answers
52 views

Ring isomorphism given by $\varphi: R \to \text{end}_R(M), a \mapsto \lambda_a$

Let $R$ be a commutative Ring with $1$ and $M$ a $R$-Module. $$\varphi: \begin{cases}R & \longrightarrow \text{end}_R(M) \\ a & \longmapsto \lambda_a \end{cases} $$ is a Ringisomorphism for ...
1
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1answer
38 views

Prove the identity in Ring of Integers Modulo Prime

I have many study tasks, but I do not have any example. Therefore, I do not know, how to solve these tasks. For example, I need prove, that: $\{ b \in \mathbb{Z}_{p^n} \mid b^2 =1\} = \{-1, 1 \}$, ...
0
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0answers
41 views

Definition of $K[x,y]/\langle f(x,y), g(x,y)\rangle$

Could you explain me the definition of $$K[x,y]/\langle f(x,y), g(x,y)\rangle?$$ How can we show for example that $$\mathbb{C}[x,y]/\langle x-1, y+x^2-1\rangle $$ is a field?
0
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2answers
61 views

Intersection of two flat submodules

Let $A$ be a ring, $M$ an $A$-module and $M_1,M_2$ two flat $A$-submodules of $M$. Is $M_1 \cap M_2$ a flat $A$-submodule of $M$?
3
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1answer
46 views

Characterize semisimple rings of quotients

Problem Let $K$ be a field. Characterize all polynomials $f\in K[X]$ such that $R=K[X]/\langle f\rangle$ is a semisimple ring. I know two equivalent definitions of semisimple modules but I am not so ...
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2answers
59 views

Irreducible polynomials in two and four variables

I am trying to show that the polynomials : 1) $X^2+Y^2-1$ 2) $XT-YZ$ are irreducible in $\mathbb Q[X,Y]$ and $\mathbb Q[X,Y,Z,T]$ respectively. For 1) I know that $\mathbb Q[X,Y]=(\mathbb ...
0
votes
1answer
20 views

A quick question on the subring generated by a finite set

Let $R$ be a commutative unital ring and let $r_1,\ldots,r_n \in R$. Let $S$ be the unital subring of $R$ generated by $r_1,\ldots,r_n$. Let $\varphi:\mathbb{Z}[X_1,\ldots,X_n]\to R$ be the unique ...
2
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0answers
19 views

Proof of Gauss lemma

Gauss lemma Let $R$ be a UFD and $F$ its field of quotients. Let $f=\sum_{i=0}^n a_ix^i \in R[X]$ with $a_0 \neq 0$. If $p$ and $q$ are non zero, coprime elements in $R$, such that $\dfrac{p}{q} \in ...
2
votes
1answer
48 views

Find a non-principal ideal (if there exists any) in the rings Z[x], Q[x], Q[x, y]

I know that $Q$ is a field, which makes $Q[x]$ a PID, which means there are none. I'm having trouble with the notation for ideal generators, and i know the $Z[x]$ has to do with something that looks ...
1
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1answer
33 views

Show that a polynomial f(x) over a field k is irreducible if and only if the polynomial f(x + 1) is irreducible.

I am very much unsure what definitions and formulas are relevant for this question. I've toyed around with the lemma "An element a ∈ R is a root of a polynomial f ∈ R[x] if and only if (x − a) divides ...
1
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2answers
57 views

Prove that $I$ is a maximal ideal

I have a question. To show that the ideal $I=\langle f(x)\rangle $ is a maximal ideal of $K[x]$ do I have to show that $f(x)$ is irreducible in $K[x]$? Or is there an other way to prove that $I$ is a ...
0
votes
1answer
58 views

Given $x$ and $y$ in $\mathbb{Z}[i]$, find $q$ and $r$ such that $x=qy+r$.

Find $q, r \in \mathbb{Z}[i]$ such that: $1 + 5i = (1 + 2i)q + r$ with $|r| < 2$, $1 + 5i = (2i)q + r$ with $|r| < 2$. My only train of thought is that $r = 1+0i$, $0+i$ or ...
2
votes
1answer
101 views

On Bounded Index of Nilpotency of $R[x]$ and $M_n(R)$

A ring $R$ is said to have a bounded index (of nilpotency) if there is a positive integer $n$ such that $x^n=0$ for every nilpotent $x∈R$. Can anyone give me an example of a ring $R$ which has a ...
0
votes
0answers
11 views

Smallest Subring of R that contains $S \cup \{a\}$

I need a bit of help with this question, I already have proven the first bit: Let $R$ be an abelian and unital ring and let $S$ be a subrinrg of $R$ such that $1_R \in S$. Also, let $a \in R$. show ...
1
vote
1answer
28 views

When is $r = r^{2}$ in $\mathbb{Z}/p^{l}\mathbb{Z}$?

When is $r = r^{2}$ in $\mathbb{Z}/p^{l}\mathbb{Z}$, where p is a prime number and l is a natural number? It obviously is the case for [0] and [1], but I am having difficulties proving that it's not ...
2
votes
3answers
74 views

When does a ring map $R\to S$ produce a group epimorphism $GL_n(R)\to GL_n(S)$?

Let $R$ and $S$ be rings with $1$ (not necessarily commutative) and $f:R\to S$ a ring homomorphism preserving $1$. Let $\bar{f}$ be the ring map $M_n(R)\to M_n(S)$ given by $f$ acting on the matrix ...
0
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0answers
12 views

Why $f(0)\neq 0$ where $f $ is a polynomial over the field $F_q$ and $deg(f)=m > 0$?

To construct the Residue class ring $F_q[x]/(f)$ having $q^m-1$ non-zero elements. Is it necessary for $f(0) \neq 0$? Why or why not? I have worked with different examples such as $x^3+x=f \in ...
0
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0answers
24 views

Noetherian Rings

Show that every principal left ideal ring is Noetherian. I know to be Noetherian one of the equivalent conditions is that it has to be finitely generated. Now, in a principal left ideal ring, all of ...