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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Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring

I read "Algebra: Chapter 0" by P.Aluffi. I encountered a topic where it says you can divide any polynomial in $R[x]$($R$ is any ring) by a monic polynomial(that is, a polynomial of the form $x^d + \...
4
votes
1answer
37 views

Maximal ideals of $C\big((0,1)\big)$

We know that all the maximal ideals of $C([0,1])$ is in the following form $$ M_a=\{f\in C[0,1]\ :\ f(a)=0 \}\ \text{for some } a\in [0,1] $$ But suppose that if we will replace our domain to compact ...
1
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1answer
88 views

Number of distinct prime ideals in $\mathbb {Q}[x]/\langle x^m-1 \rangle$ [closed]

Number of prime ideals in quotient ring obtained by $\mathbb {Q}[x]/\langle x^m-1 \rangle$ is ...?
1
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1answer
24 views

Essential Prime Ideal

I search for an example of a commutative ring $R$ with unity having a prime ideal $P$ and some element $r\in R$ such that the annihilator of $r$ is both contained in $P$ and essential in $R$. By ...
0
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1answer
62 views

About commutative ring with identity

Let $R$ be an infinite commutative ring. Which of following options is false? Center of $M_2(R×R)$ is nontrivial. $ M_2(R×R) \cong M_2(R)×M_2(R)$ The number of units in $M_2(R ×R)$ is infinite. The ...
-1
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1answer
30 views

Ideal which is maximal with respect to the property that it does not contain the set S

Question Let $R$ be a ring and let $S$ be a subset of $R$. Prove that there exists an ideal that is maximal by inclusion with respect to the property that it does not contain the set $S$. I ...
9
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3answers
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Defining binomial coefficients over rings in general

I'm looking over my notes, and we proved that the binomial theorem $$(a + b)^n = \sum_{k = 0}^{n} {n \choose k}a^k b^{n-k}$$ holds on all commutative rings. However, I am confused on how we are ...
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1answer
24 views

UFDs and reduced quotients

Let $A$ be a UFD and let $x\in A$ an element. I don't understand why the following claim is true: The quotient $A/(x)$ is reduced if and only if $x$ is a product of distinct primes. Can you suggest ...
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2answers
33 views

Why Is $I$ A Maximal Ideal If $R/I$ Is A Field?

How can I prove that if $R/I$ is a field then $I$ is a maximal ideal? In my book it says that this is a corollary of the following theorem: If $φ:\mathbb{F}→S$ is a non trivial homomorphism and $\...
2
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1answer
27 views

For $\mathbb{Z}[\sqrt{-10}]$ and an ideal $I=(2,\sqrt{-10})$, find $n>0$ s.t. $I^n$ is a principal ideal.

I can show that $I$ itself is not a principal ideal by using the multiplicative norm $N(a+b\sqrt{-10}) = a^2 + 10 b^2$ using the following: Suppose that $I=(\alpha )$. Then $N(\alpha) | 2$ and $N(\...
2
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1answer
45 views

Why is it necessary to show that the set R is closed under multiplication in order to prove that (R,+,*) is a Ring.

I am currently reading John B Fraleigh's book A First Course in Abstract Algebra. The author in chapter 23, insists on the point that it is necessary for one to show that the R is closed under ...
5
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2answers
55 views

How to construct rings with a given class number?

Hi I was learning about class number and I was wondering if it is known how to construct rings for any specific class number.
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1answer
35 views

Conjecture about the elements of quotients of polynomial rings over commutative rings by principal ideals

Let $R$ be a commutative ring with unity. For any polynomial $p(x) \in R[x]$ is there an element $a$ of $R[x]/ \langle p(x) \rangle$, $a \not \in R$ such that $a^n \in R$, where $n$ is the degree of $...
3
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1answer
72 views

Example of non-principal ideal of the quotient ring $R[x,y,z]/(xyz-1)$

Suppose $R$ is an integral domain, $R[x,y,z]$ is polynomial ring over $R$, and $$Q=R[x,y,z]/(xyz-1)$$ is a quotient ring. How to prove, that $Q$ is not principal ideal ring? I was trying to compose an ...
1
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1answer
34 views

Direct sum decompositions over rings.

Let $R$ be a ring and let $I$ be one of its nonzero left-ideals. Is it true that $R \cong I \oplus R\big/I$? Notice that we can consider $R$ to be an left $R$ module and $I$ one of its $R$-submodules....
2
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1answer
33 views

Are Noetherian Rings Goldie Rings?

We know that a ring $R$ is said to be Right-Goldie if $R$, as a right $R$ module, satisfies: (i) $R$ has finite uniform dimension; (ii) every ascending chain of right annihilators of $R$ terminate. ...
0
votes
1answer
35 views

$p_1q_1q_2$ and $p_1p_2q_1$ do not have a greatest common divisor

Question Let $p_1,p_2,q_1,q_2$ be irreducible elements in integral domain $R$ such that none are associates to any of the others and $p_1p_2=q_1q_2$. Prove that $p_1q_1q_2$ and $p_1p_2q_1$ do ...
0
votes
1answer
18 views

Some equivalences for Ideals of the ring of real valued functions

Let $I_{N}=\{f\in\Omega(\Bbb R,\Bbb R)|\forall x \in N\subset \Bbb R: f(x)=0\}$, where $\Omega(\Bbb R,\Bbb R)=\{f:\Bbb R \to \Bbb R|f\space is \space a\space function\}$, then the following statements ...
0
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1answer
31 views

quotient of P.I.D by a prime power a P.I.D? [closed]

If $R$ is a P.I.D. and $p\in R$ is prime, is it the case that $R/<p^k>$ will be a P.I.D for all k? If so how would one show this?
1
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1answer
10 views

if an element $q_1$ of a principal ideal domain is irreducible, then the ideal $q_1 A$ is a maximal ideal

In my lecture, my professor wrote within a proof: $q_1$ in the principal ideal domain $A$ is irreducible, therefore, the ideal $q_1A$ is a maximal ideal. I don't understand how that's true. I tried ...
1
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1answer
24 views

Structure of the proof of the fundamental theorem of arithmetics

Fundamental theorem of arithmetics: A principal ideal domain is factorial. i.e: Any non zero element of a principal ideal domain can be decomposed in a unique product of irreductibles. The structure ...
3
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0answers
59 views

Show that $(\mathbb{Z}\oplus\mathbb{Z})[x]$ is not isomorphic to $\mathbb{Z}[x]\oplus\mathbb{Z}[x] $ . [duplicate]

Problem says: Show that $(\mathbb{Z}\oplus\mathbb{Z})[x]$ is not isomorphic to $\mathbb{Z}[x]\oplus\mathbb{Z}[x]$. But consider the map $\varphi:(\mathbb{Z}\oplus\mathbb{Z})[x]\rightarrow\...
1
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1answer
108 views

Prove or Disprove: there is only one ring homomorphism $f:\mathbb{R} \rightarrow \mathbb{C}$

I searched a lot but I couldn't solve my problem! I know that $$ f(1) = f(1.1)= f(1).f(1) \Longrightarrow f(1) = 0 \quad or \quad f(1) = 1 $$ I know that if we suppose that $f(1) = 0$ then $f$ is ...
1
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1answer
40 views

Question About Irreductibility of an element in a ring

I $A$ is a principal ideal domain and let N the set of non zero elements, that do not have an inverse and that cannot be decomposed in a product of irreductibles. If $a \in N$ Let's create the ...
0
votes
1answer
14 views

Proof of a proposition regarding rings and ideals

I am looking for a proof of the following proposition: If A is a principal ideal domain, let the following increasing sequence of Ideals of A: $I_1⊆I_2⊆I_3⊆...⊆I_m⊆...$ Then there exists $n \geq 1$ ...
3
votes
1answer
24 views

What is known about finite-dimensional non-semisimple associative algebras over $\Bbb C$?

Artin-Wedderburn says the semisimple ones are sums of matrix algebras, but what about the non-semisimple ones? What are some examples?
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0answers
29 views

Finite module over Noetherian ring faithfully flat?

If I have Noetherian rings $B=A^G\subset A$ (for some action of finite group $G$, maybe not relevant) and $A$ is finite as $B$-module. Is it always true that $A$ is faithfully flat over $B$? EDIT: ...
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0answers
34 views

Is there a term for the dimension of the annihilator of an element of an algebra?

Let $\mathcal{A}$ be a finite dimensional $R$-algebra, and for $x \in \mathcal{A}$ consider $\mathrm{Ann}(x) = \{ c \in \mathcal{A} \mid cx = 0 \}$. Is there a term for the dimension of this subspace? ...
3
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1answer
33 views

Are there any finite dimensional associative and commutative algebras which are reduced, but not semisimple?

In other words, I am looking for a finite dimensional associative and commutative algebra with non-trivial Jacobson radical, but trivial nilradical. Does such a finite dimensional associative and ...
4
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0answers
80 views

How do ring theorists think about square roots?

Let $R$ denote a commutative ring. Then it seems to me that we can adjoin to $R$ a square-root of $4$ as follows: $$R[\sqrt{4}] = R[x]/(x^2-4)$$ This defines a functor $\mathbf{CRing} \rightarrow \...
2
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2answers
52 views

Given two polynomials $f$ and $g \in \mathbb{Q}[X]$, prove that $(f) + (g) = (h)$ and $(f)\bigcap(g) = (k)$

Given two polynomials $f(X) = 3X^2 + 7X - 6$ and $g(X) = 2X^2 + 5X - 3 \in \mathbb{Q}[X]$, prove that there exist $(h)$ and $(k) \in \mathbb{Q}[X]$ such that $(f) + (g) = (h)$ and $(f)\bigcap(g) = (k)$...
0
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2answers
40 views

How to prove that the image of a prime ideal is also a prime ideal

If $f:A\rightarrow R$ be a ring homomorphism, where $A$ and $R$ are commutative rings. If $f$ is surjective and $P$ is a prime ideal in $A$, how to prove that $f(P)$ is a prime ideal in $R$? This ...
1
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1answer
23 views

Describe the Kernel of the map from this polynomial ring

Let $\mathbb R[x]$ denote the ring of all polynomials with real coefficients. The mapping $f(x)\rightarrow f(1)$ is a ring homomorphism from $\mathbb R[x]$ onto $\mathbb R$. Question: Describe ...
2
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3answers
45 views

How can I prove that $\mathbb Z_3[i]\cong \mathbb Z_3[x]/\langle x^2+1\rangle$?

Question: Let $\mathbb Z_{3}\left [ i \right ]=\left \{ a+bi\mid a,b \in \mathbb{Z}_{3} \right \}.$ Show that the field $\mathbb Z_{3}\left [ i \right ]$ is ring isomorphic to the field $\...
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2answers
84 views

Give an example of a commutative von Neumann regular ring which is not a product of fields

One knows that every commutative von Neumann regular ring with a finite Boolean algebra of idempotents is a product of fields. Give an example of a commutative von Neumann regular ring which is ...
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0answers
33 views

the field of fractions of a domain $R$ does not have a projective cover if $R$ is not a field.

Let $R$ be an integer domain (so it commutes) that is not a field. Let $K$ be its field of fractions (also called localization). Prove that $K$ does not have a projective cover ( naturally viewed as ...
0
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2answers
31 views

Showing Quotient ring is a field using maximal Ideal

Question: Show that $R\left [ x \right ]/\left \langle x^{2}+1 \right \rangle$ is a field. Recall: Theorem: Let R be a commutative ring R with unity. Let I be a proper Ideal of a ring R. ...
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1answer
8 views

Verify size of factor ring

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$ and let I be the subset of R consisting of matrices with even ...
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1answer
30 views

Verifying an isomorphism between coordinate rings

I am trying to convince myself of an isomorphism between: $$k[x,y,z]/(x^2-yz,z-1) \rightarrow k[t]$$ In trying to show that these rings are isomorphic, I have constructed a map sending: $x \...
0
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1answer
35 views

Kernel of ring homomorphism $k[X,Y] \rightarrow k[t^2, t^3]$ [duplicate]

Let $k$ be a field, $f: k[X,Y] \rightarrow k[t^2, t^3]$, $X \mapsto t^3, Y\mapsto t^2$. I would like to verify that $\ker f = (X^2 - Y^3)$. It is easy to see that $X^2 - Y^3 \in \ker f$ and ...
2
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1answer
21 views

Any ideal of a field $F$ is $0$ or $F$ itself

Prove that the only ideals of a field are $\left\{ 0 \right\}$ and the field itself. Let $F$ be a field and $I$ be an Ideal of $F$. Let $0 \ne x \in I$. Since $I$ is an Ideal of $F$, it is true ...
4
votes
1answer
45 views

Ring action on another ring?

So a module over a commutative ring $ R$ is an abelian group $G$ equipped with an action given by the product $R\times G\rightarrow G$ that satisfies a few conditions. What if $G$ itself is a ring? Is ...
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0answers
35 views

What does $F[x]$ mean?

Lemma: $F$ is a field only if $F\left [ x \right ]$ is a Principal Ideal Domain. This is a theorem from Ring; divisibility of integral domain. What does $F\left [ x \right ]$ mean?
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2answers
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Is a sub-algebra of a semisimple algebra nescesarily isomorphic to a subset of its direct product decomposition via Wedderburn's classification?

Wedderburn's classification of semisimple algebras tells us that any semisimple algebra $A$ is isomorphic to a finite direct product of matrix algebras over division algebras, say $A \cong M_{n_1}(D_1)...
2
votes
1answer
60 views

Is there a way to characterize the prime ideals in $\mathbb{R}[x_1,x_2, \dots , x_n]$?

I'm studying algebras which can be formed by the quotient of principal ideals in $\mathbb{R}[x_1, \dots , x_n]$, and thus would like to be able to determine which of said principal ideals are maximal, ...
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0answers
10 views

A modularity condition

Let $R\subseteq S$ be rings with unity and $X$ ,$Y$ be subsets of $S$ with $X$ an ideal. If $S=X+Y$ what conditions should be held to infer the equality $R=(X∩R)+(Y∩R)$? I think that if we ...
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2answers
56 views

Prove that $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(i \sqrt{5})$ are not isomorphic.

The question is : Prove that $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(i \sqrt{5})$ are not isomorphic (I'm talking about ring isomorphism). What I have done : suppose there is an isomorphism $f:\...
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0answers
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Prove that $M_p$ is a ideal of $\mathbb Z/(p)[x]$ and $\mathbb Z[x]/M$ is isomorphic to $\mathbb Z/(p)[x]/Mp$.

Let $M_p$ = $\gamma (M)$, the image of $M$ ($M$ is a maximal ideal of $\mathbb Z [x]$) in $(\mathbb Z/(p))[x]$, where $\gamma$: Z[x] --> Zp[x] is the morphism such that $\gamma (\sum_i a_ix^i)=\sum_i [...
1
vote
1answer
21 views

If $a \in A$ is not a prime number, then $A/aA$ is not an integral domain

If $a \in A$ is not a prime number, then $A/aA$ is not an integral domain: proof $a$ not prime, therefore: $a \mid bc$ and $a\nmid b$ and $a\nmid c$ Therefore $b+aA \neq aA$ and $c+aA \neq aA$ ...