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Questions tagged [ring-theory]

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
26 views

Maximal Ideal in a polynomial ring

Let $G$ be a field, $c \in G$, and let $H = \{(x - c)k, k \in G[X]\}$ be an ideal of the polynomial ring $G[X]$. How to show that $H$ is a maximal ideal? Ideas: I know that $H$ is maximal if and ...
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2answers
44 views

What is $\mathbb{R}[x]$ quotiented by a polynomial $f(x) \in \mathbb{R}[x]$ isomorphic to?

By the CRT, we have that: $$\mathbb{R}[x]/(x^2-2) \simeq \mathbb{R}[x]/(x-\sqrt{2}) \times \mathbb{R}[x]/(x+ \sqrt{2}) \simeq \mathbb{R} \times \mathbb{R}.$$ By considering the evaluation map $\...
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1answer
28 views

Proving associativity of “xor” in any complemented distributive lattice

If, in a Boolean algebra $(X,\vee,\wedge,0,1,')$ (I want to say a complemented distributive lattice), we define $x+y=(x'\wedge y)\vee(x\wedge y')$, is there an elegant way to prove that $(x+y)+z=x+(y+...
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1answer
26 views

$a^{50}x'+b^{20}y'+c^{15}z'=1$ for $a,b,c,x',y',z'\in R$

Let $R$ be a commutative ring with unity. If for some $a,b,c\in R$, we have $x,y,z\in R$ such that $ax+by+cz=1$, then does it imply that $\exists x',y',z'\in R$ such that $a^{50}x'+b^{20}y'+c^{15}z'=1$...
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1answer
28 views

Question regarding the image of the unity $e$ of the ring $R$.

I was reading Ring Homomorphism . $\phi : R \to R'$ is a ring homomorphism and $e , e'$ are the unities of $R $ and $R'$ respectively. I understood that $\phi (e) $ may not be unity of $R'$. I ...
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0answers
31 views

UFD: existence of an infinite factorization

The definition of UFD requires that each non-unit, non-zero element to have a finite, unique factorization of irreducibles. But is it possible for a non-unit, non-zero element to also be a product of ...
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2answers
36 views

Show $f:\mathbb{Z}_6 \rightarrow \mathbb{Z}_3$ is a homomorphism

I am currently studying for an abstract algebra final exam. I am trying to disprove the statement "Consider the homomorphism $f: R \rightarrow S$ where R and S are rings. Prove/Disprove: If $a \in R$ ...
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1answer
27 views

Let R be a commutative ring, and I, J denote two ideals in R such that I + J = R. Is it true that IJ = I ∩ J? [duplicate]

So far I have that if I + J = R, then 1 ∈ I and/or 1 ∈ J. Then I = R and/or J = R. If both I = J = R, then IJ = I ∩ J = R must be true because we already know that multiplying every ...
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1answer
32 views

$\mathbb {Z}_{84}/(7) \cong \mathbb {Z}_{7}$

Prove $\mathbb {Z}_{84}/(7) \cong \mathbb {Z}_{7}$ using each of the three isomorphism theorems for rings. For the first isomorphism theorem I defined a homomorphism $\phi: \mathbb {Z}_{84} \to \...
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0answers
27 views

Connection between properties of a ring and its quotient rings

Let $R$ be a $k$-algebra, $k$ a field, and $0\neq I \neq R$ a two-sided ideal of $R$. Denote by P a property of rings. One says that $R$ is 'just P', if $R$ does not satisfy property P but $R/I$ ...
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2answers
20 views

Let R be a domain with 1. Show that if aR = bR, then au = b for some unit u ∈ R. [duplicate]

Let $R$ be a domain with $1$. Show that if $aR = bR$, then $au = b$ for some unit $u ∈ R$. Any hint on how to start this proof?
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0answers
26 views

Specific question on lcm and gcd of rings.

I can't prove this statement: Let $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ be non zero elements of an integral domain $R$ such that $a_1b_1=a_2b_2=\cdots=a_nb_n=x$ If $gcd(ra_1,ra_2,...,ra_n)$ exists ...
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2answers
39 views

Factorization in prime elements of $\mathbb{Z}\left[\sqrt{p}\right]$ for a prime number $p$

I'm having troubles with the following problem: Let $p$ be a prime number in $\mathbb{Z}$, and $\alpha\in\mathbb{Z}\left[\sqrt{p}\right]$ which is not a unit. Prove that $\alpha$ have a factorization ...
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0answers
16 views

Does $\DeclareMathOperator{\len}{length}\DeclareMathOperator{\rk}{rank}\len(M/xM) \leq \rk(M) \cdot \len(R/(x))$ hold over non-integral rings $R$?

$\DeclareMathOperator{\len}{length} \DeclareMathOperator{\rk}{rank}$In Eisenbud's book Commutative Algebra with a View towards Algebraic Geometry he says: The basic result of this section expresses ...
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2answers
35 views

Injective Homomorphism from $\mathbb{R}\times\mathbb{R}$ to the ring of Continuous functions

Does there exist an injective ring homomorphism from the ring $\mathbb{R}\times\mathbb{R}$ to the ring of continuous functions over $\mathbb{R}$? I know that $\mathbb{R}\times\mathbb{R}$ is a field. ...
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0answers
21 views

Direct Product cancellation in Hopfian rings

A ring $R$ is called to be 'Hopfian' if every ring homomorphism of $R$ onto $R$ is an automorphism of $R$ Question: Given $R$, $S$, $T$, Hopfian rings and $$R \times S \cong T \times S$$ implies ...
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0answers
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Showing that $(I+J)/(I\cap J)\cong (I+J)/I\times (I+J)/J$.

Show that $(I+J)/(I\cap J)\cong (I+J)/I\times (I+J)/J$, where $I,J$ are ideals in a commutative ring $R$ with identity. What I did was first show that $\phi: I+J\to (I+J)/I\times (I+J)/J$ defined by $...
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1answer
23 views

Showing that $M_n(R[x]) \cong (M_n(R)[x]$

I'm trying to show that $M_n(R[x]) \cong (M_n(R)[x]$ so I consider the mapping that sends an element $A \in M_n(R[x])$ to the polynomial whose coefficients are matrices in which the entries of those ...
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0answers
35 views

Ideals in $Z_{18}$

Hello I am trying to find the ideals of Ideals in $Z_{18}$ I got confused so I looked at the back of the book which had $<2>$ & $<3>$ as the answer and said they were both maximal ...
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2answers
44 views

Axioms of algebraic structure - ring

If in the definition of ring $(R,+,\times$) we insist that it has unit element $1$. Then we can show that addition $(+)$ is commutative operation. However, most of the proof which I've seen in MSE use ...
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0answers
37 views

Some counterexamples in basic ring theory

Give an example if possible, and if not possible explain why not. a) A subring of a PID that is not PID. b) A PID that is a subring of a non-PID. c) A subring of a PID that is not UFD. My approach:...
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0answers
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About irreducible representations over the polynomial ring $k[x]$

From Example 2.3.14 (2) here page 21: Let $A = k[x]$. Since this algebra is commutative, the irreducible representations of $A$ are its 1-dimensional representations. They are defined by a single ...
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0answers
24 views

Explain why J is a prime Ideal in Z[x] [duplicate]

I am trying to prove that the ideal $J=<x+1>$ is prime in the ring $\mathbb{Z}[x]$. I know that if the generator is prime, then the ring modulo the generator is an integral domain. I can show ...
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0answers
29 views

Ring-like strusture with non associative addition

There is this structure i found which is the set of continuous maps from [-1, 1]^n into itself, endowed with a "sum" which is the pointwise sum of two functions divided by 2, and a "product" which is ...
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0answers
9 views

Why is the one-variable ring of polynomials with real coefficients a principal ideal domain? [duplicate]

I'm new to Ring Theory and I'd like to check if I'm on the correct track with this. Wolfram defines a PID as "an integral domain in which every proper ideal can be generated by a single element". As ...
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1answer
42 views

Is $R$ an algebra or not?

In the book Quasi Frobenius Rinngs- Nicholson and Yousif, Example 2.5 gives a ring as follows: Let F be a field and assume that $a→ \bar{a}$ is an isomorphism $F → \bar{F} ⊆ F$, where the subfield $\...
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1answer
19 views

All ideals in $\mathbb{Z}[\frac{1}{2}]$ are finitely-generated

The question comes from a problem, which asks to find all prime and maximal ideals in $\mathbb{Z}[\frac{1}{2}] := \{\frac{a}{2^k} : k\in\mathbb{Z} , a\in 2\mathbb{Z}+1 \} $. Well I solved it by ...
1
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1answer
58 views

What does $φ(a) = a$ mean in this statement?

Let $F$ be a field and let $φ:F[x] \to F[x]$ be an isomorphism such that $φ(a)=a $ for every $a$ in $F$. Prove that $f(x)$ is irreducible in $F[x]$ if and only if $φ(f(x))$ is. [Hint: First prove that ...
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1answer
54 views

Integral Closure, Galois extension,and Dedekind Domain

Let $A$: Dedekind domain, $K$: $\operatorname{Frac}(A)$, $B$: Dedekind domain with $A \subset B$, $L$: $\operatorname{Frac}(B)$ Let $L/K$: galois extension with galois group: $G$. $B^G=\{b \in B \...
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1answer
41 views

Determining ideals, isomorphic rings of $\Bbb C[x, y]/(y^2 - x^3)$?

I've been having a substantial amount of trouble trying to understand the workings of $\Bbb C[x, y]$ mod... anything really. I figure this particular example is a good one to ask here because I ...
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1answer
45 views

Showing an isomorphism between two quadratic rings

Let $a,b$ be squarefree integers and set $R = \mathbb{Z}[\sqrt{a}]$ and $S = \mathbb{Z}[\sqrt{b}]$. Prove that 1) There is an isomorphism of abelian groups $(R,+) \cong (S,+)$. Let $\varphi : R \to ...
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2answers
54 views

Ideals generated by two elements in $\mathbb{Z}[x]$

Consider the following ideal $(2+x,x^2+5)$ in $\mathbb{Z}[x]$. Then I showed that $(2+x,x^2+5)=(9,2+x)$. But I am not able to do the same for ideals $(1-4x,x^2+5)$ and $(1+2x,x^2+5)$. Is there some ...
1
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1answer
11 views

Let $K$ be the ring of all real functions and let $f \in K$. Suppose that f is not a zero-divisor. Prove that $f \in U(K)$. [on hold]

Let K be the ring of all real functions and let $f \in K$. Suppose that $f$ is not a zero-divisor. Prove that $f \in U(K)$.
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3answers
34 views

Prime ideals in certain quadratic ring

Let's consider the quadratic ring $\mathbb{Z}[\sqrt{-5}]$ and the principal ideals $(29)$ and $(11)$. Tell whether or not these ideals are prime. My approach: In order to solve this theorem I am ...
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2answers
34 views

Prove $(0) = (x)\cap (xz^{n-1} + \lambda y^n)$ in $R=\frac{k[x,y,z]}{(x^2,xy)}$

Studying for my algebra final and doing some practice problems, and I can't seem to understand this one... Full problem: Let $k$ be a field, and $R=\frac{k[x,y,z]}{(x^2,xy)}$. For $n\in\mathbb{N}, \...
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0answers
20 views

On the number of multiplicative sub-monoids of integers mod $n$

Let $n \geq 2$ be given. Is there a formula describing the number of multiplicative closed subsets of the ring $\mathbb{Z}_n$ ?
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1answer
41 views

Does every integral domain come from a quotient?

Let $A$ be conmutative ring with identity and $\mathfrak{p}, \mathfrak{m}$ ideals. Then $$\begin{array}{ll} \mathfrak{p}\text{ is a prime}\iff A/\mathfrak{p}\text{ is an integral domain}\\ \mathfrak{m}...
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2answers
31 views

How the following multiplication table is solved ( related to $F_2[X]/f(x)$ )

$F_2$ is polynomial field of group of integer modulo $2.f(x)$ is $x^2 + x + 1$. I didn't got how the multiplication is happening in the table.I referred to many sources related to this topic but ...
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2answers
43 views

Let $R \subseteq S$ be two local PIDs with the same field of fractions, then $R=S$.

Let $R$ and $S$ be two local principal ideal domains with the same field of fractions $K$. I want to show that if $R\subseteq S$ then $R=S$. I will denote as $\mathfrak{m}_R=(m_R)$ and $\mathfrak{m}...
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1answer
99 views

If the square of every element of a ring is in the center, must the ring be commutative?

Let $R$ be a ring with identity such that the square of any element belongs to the center of $R$. Is it necessary true that $R$ is commutative? (I can show that for any $x,y\in R$, $2(xy-yx) =0 $ but ...
3
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2answers
38 views

Construct a ring containing $16$ element where EVERY element $r\neq 0$,$1$ is a zero divisor

This is a practice question to prepare me for my final exam in abstract algebra. Construct a ring containing $16$ elements where EVERY element $r\neq 0$,$1$ is a zero divisor I'm having a hard time ...
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2answers
43 views

If $R$ is a noncommutative ring such that $\exists a,b \in R$ s.t. $ab=1$ and $ba \neq 1$ then $R$ is infinite [duplicate]

If $R$ is a noncommutative ring and there exist $a,b\in R$ such that $ab=1$ and $ba\neq 1$, theN $R$ is infinite. I need some help with this one, been playing around with elementary algebra but am ...
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0answers
27 views

Prime ideals in quadratic ring $\mathbb{Z}[\sqrt{-5}]$

Consider quadratic ring $\mathbb{Z}[\sqrt{-5}]$. For each of the following elements tell whether or not the principal ideal $\langle x\rangle$ generated by $x$ is a prime ideal. $x=29,11.$ My ...
3
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1answer
41 views

If $J$ is an ideal of $R$ that is maximal in the set of ideals of $R$ that annihilate elements of $R/I$, then $J$ is a prime ideal of $R$.

Let $R$ be a ring and let $I$ be an ideal of $R$. Show that if $J$ is an ideal of $R$ that is maximal in the set of ideals of $R$ that annihilate elements of $R/I$, then $J$ is a prime ideal of $R$. ...
4
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2answers
53 views

$\bigcap_{n\in\mathbb{N}} I^n = (0)$ if and only if no zero divisor of $R$ is of the form $1-z$ with $z\in I$.

Full problem, suppose that $R$ is a commutative Noetherian ring and $I$ is an ideal of $R$. We wish to prove that $$\bigcap_{n=1}^{\infty} I^n=(0)$$ if and only if no zerodivisor of $R$ is of the ...
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0answers
33 views

Examples of ring where doesn't exist lcm and gcd

My question is: "Examples of ring that doesn't exist lcm and gcd of any elements" The ring preferely has to be commutative and unitary or olny unitary( as matrix ring). I woul use these two ...
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0answers
18 views

Invariant basis number for Rings.

Let $R$ be a ring with identity. Let $M_{n}(R)$ be the ring of $n$ by $n$ matrices with entries in $R$. Prove that $R$ has IBN iff $M_{n}(R)$ has IBN. I was given this problem. I thought a little ...
1
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1answer
61 views

Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous?

Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous ? Why $v(a) \leq v(ab)$ is not needed? How we can deduce from the first one?
6
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2answers
96 views

Non unique factorization of integer valued polynomials

Is there a nice example of a polynomial with non unique factorization in the subring of $\mathbb Q[X,Y]$ of polynomials that defines functions $\mathbb Z^2\to\mathbb Z$? I don't think this subring ...
0
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1answer
26 views

If $g = gcd(a,b)$ prove (a,b)=(g). Furthermore, if $k = lcm(a,b)$ prove that $(a)\cap(b) = (k)$

$a,b \in \mathbb{Z}$. $(a,b),\hspace{0.4mm}(g)$ and $(k)$ are principle ideals I'm new to this kind of problems, so I don't even know how to start it. Some help would be appreciated.