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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Is <x,5> a maximal ideal in Z[x]?

Here $<x,5>$ is the ideal generated by $x$ and $5$ in $\mathbb Z[x]$ that is the polynomial ring in $\mathbb Z.$ How should I approach this question ?
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0answers
27 views

Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.

Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. Okay so here is the proof I came up with PLEASE be as ...
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2answers
33 views

Let $R$ be a finite ring with unity. Prove that $x$ is a LZD $\iff$ x is a RZD

Let $R$ be a $finite$ ring with unity. Let $x \in R$. Prove that $x$ is a Left Zero Divisor $\iff$ x is a Right Zero Divisor. My attempt Suppose $x$ is a LZD. Then, $\exists y \in R$ such that $xy = ...
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0answers
36 views

A ring with a left cancellable element and a right identity always has an identity.

Let $R$ be a ring with $a, e \in R$ such that $a$ is not a left zero-divisor and $be=b, \forall b \in R$. Prove that $R$ has an identity. My attempt Let, $aeb = ab \Rightarrow aeb - ab = 0 ...
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0answers
35 views

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the ...
2
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1answer
30 views

To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $.

To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $. Thus there exists an ideal $J$ of $ \Bbb Z \times \Bbb Z $ such that $I ...
2
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1answer
31 views

Ring Homomorphism from $\mathbb{Z}_m$ to $\mathbb{Z}_n$

Suppose $R$ is a ring homomorphism from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ , prove that if $R(1) = a$ then $(a^2)=a$. Also show, its converse is not true. The first part goes like this : $R(1) = a , ...
2
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1answer
34 views

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely in the form $a(x) + (p(x))$ where $\text{deg}(a) < \text{deg}(p)$ this is a homework problem and I'm stuck, here is my ...
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2answers
32 views

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. [duplicate]

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is a ideal.(I have done it) But how to show that it is maximal?
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3answers
31 views

Elements of $\mathbb{Z}/(n)$

Let $(n) = \{ \lambda n | \lambda \in \mathbb{Z} \}$. In my book it has shown that every element in $\mathbb{Z}/(n)$ can be expressed uniquely in the form $r + (n)$ where $0 \leq r \leq n-1$ now I ...
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1answer
13 views

Commutative rings and ideals, showing a map is well defined

Let $R$ be a commutative ring with an ideal $I$. The additive group $R/I$ is the set of cosets of $I$ with respect to addition in $R$. Let $\cdot : R/I \times R/I \to R/I$ be defined by ...
2
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1answer
16 views

Proving that a Left semisimple ring $R$ is both left noetherian and left artinian

Prove that a left semi-simple ring $R$ is both left noetherian and left-artinian. I am following the proof given in pg 27,A first course in non-commutative rings (T.Y.Lam). Its strategy is to show ...
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2answers
43 views

Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$.

Let $I = \{a +ib \in \Bbb Z[i] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is an ideal of $\Bbb Z[i]$. Is $I$ a maximal ideal? And to find the numbers of elements of the quotient ring $\Bbb ...
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1answer
38 views

An approach to proving that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$

I have to prove that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$. My approach: Let us consider $t^2$ and $t^3$ as separate variables $x$ and $y$. The relations that hold for them ...
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1answer
24 views

Unique factorization consequence

Let R be principal ideal domain and $p$ prime element and $b \in R$, $b\neq 0$, $E=R/(pb)$ module over $R$ and let $E_p$ be submodule consisting of elements with exponent $p$ (there exist positive ...
3
votes
1answer
22 views

If C is a chain of non-principal ideals and the union of the ideals of the chain contains a generator, why is the union then principal

If C is a chain of non-principal ideals and the union of the ideals of the chain contains a generator why is the union then principal. I understand this is a contradiction already and seems painfully ...
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0answers
20 views

direct sum of modules is isomorphic to the direct sum permuting indices? [on hold]

the primary for the exercise idea is to use the universal property of the external direct sum of modules.
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23 views

Verification: Let $R$ be a unique factorization domain and let $a,b \in R$ be nonzero, non-unit elements. Prove that $a$ and $b$ have a LCM

Let $R$ be a unique factorization domain and let $a,b \in R$ be nonzero, non-unit elements. Prove that $a$ and $b$ have a least common multiple, and describe one such multiple in terms of the ...
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2answers
22 views

The mapping $\theta : S^{-1}R \rightarrow (\pi(S))^{-1}(R/I)$ is a well-defined ring epimorphism.

I'm working on this problem for a homework assignment. Note that $R$ is a commutative ring with unity, $I$ is an ideal of $R$, and $\pi : R \to R/I$ is the canonical projection given by $\pi(r)=r+I$. ...
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2answers
20 views

Show that if $F$ is a field, then $<x>$ is maximal in $F[x]$. Also, show that $F[x]$ is not local.

See statement above. So far I have the following: Assume that $<x>$ is not maximal. Then $ <x> \subset <f(x)> \neq F[x]$. This means that $x = f(x) g(x)$. Since $x$ is ...
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2answers
36 views

Prove that $\mathbb{Z}_{mn}$ has atleast four idempotent elements. [duplicate]

Suppose $m, n > 1$ are positive integers which are relatively prime. Prove that $\mathbb{Z}_{mn}$ has atleast four idempotent elements. Two of them are $[0], [1]$, how will I find the other two?
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0answers
91 views

Cardinomials: Like cardinalities, but polynomial valued

I want to see if this notion is known (or if it makes sense). Let $F$ be a field. Let $A$ be a finite dimensional commutative unital algebra over $F$. Let $X_1$, $X_2 \in A$ etc. be such that their ...
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2answers
32 views

Is it true that the order of any quotient ring $\mathbb Z[i]/\langle a+ib \rangle $ is $a^2+b^2$ ? (where not both $a,b$ are zero)

Is it true that the order of any quotient ring $\mathbb Z[i]/\langle a+ib \rangle $ is $a^2+b^2$ ? ( I know it is atmost finite ) Please help . Thanks in advance .
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1answer
24 views

On non-constant multiplicative norms on integral domain and when does the absolute value of the norm is unity implies the element is unit?

Consider $\mathbb Z[\sqrt {d}]$, where $d$ is any non - square integer, define $$N(a + \sqrt d b) = a^2 - db^2 = (a + \sqrt d b)(a - \sqrt d b)$$ as $\mathbb Z \subseteq \mathbb Z[\sqrt {d}]$, so from ...
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1answer
25 views

Proof of that in an integral domain, every prime element is irreducible.

I would like to prove that in an integral domain $R$, every prime element $p$ is irreducible. I understand the case where $p = ab$ but the textbooks I have read do not address the case where $p \neq ...
3
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1answer
33 views

let $G=\mathbb{Q}^*$ and $\varphi: G \to G$ where $\varphi$ interchanges 2,3 in the prime power factorization. Prove it is a group isomorphism

let $(G, \cdot)=(\mathbb{Q}^x, \cdot) = (\{\frac{p}{q}\mid\frac{p}{q} \neq 0\}, \cdot)$ and $\varphi: G \to G$ where $\varphi$ interchanges 2,3 in the prime power factorization and $\varphi$ is ...
3
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1answer
35 views

Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$.

Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$. We consider an ideal $J$ such that $I \subset J\subset\Bbb Z[i] $. So there exists an element $p \in J$ but ...
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2answers
27 views

Suppose $E$ is the quotient field of $D$ then find the relation between $D[x]$ and $E[x]$.

Let $D$ be an integral domain, then $D[x]$ is an integral domain and find its quotient field. Suppose $E$ is the quotient field of $D$. Then find the relation between $D[x]$ and $E[x]$. I have ...
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0answers
33 views

Isomorphism of direct product of quotient rings

How do I show that with $n = p^aq^b$ with $p,q$ distinct primes and $a,b \geq 1$ that $$\mathbb Z/n\mathbb Z \cong (\mathbb Z/p^a\mathbb Z) \times(\mathbb Z/q^b\mathbb Z)?$$ I am told that Bezout's ...
3
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1answer
53 views

Is every well ordered commutative nontrivial ring with identity an well ordered integral domain?

$\mathbb Z$ is up to ring isomorphism the only well ordered domain, that is, $\mathbb Z$ is a integral domain and every nonempty subset of $\{n \in \mathbb Z: n\geq 0\}$ has a least element. But what ...
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2answers
29 views

How to determine if this is a principal ideal domain?

Consider $\mathbb{Z[\sqrt{-5}]}=a+b\sqrt{-5}$ where $a,b \in \mathbb{Z}$. My understanding is that an integral domain is a PID if every ideal in the ring is principal. For the above example, this I ...
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2answers
54 views

Show that $a(-1) = (-1)a = -a $.

In a ring $R$ with identity 1, show that $$a(-1) = (-1)a = -a \qquad\forall\, a \in R$$ I have started with $a + (-a) = 0$ but cant proceed from here.
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1answer
30 views

Q has no maximal subgroups.

Theorem: If $R$ is a ring with 1 and $I$ is a ideal in $R$ such that $I \neq R$, then there is maximal ideal $M$ of the same kind as $I$ such that $I\subseteq M$. Note:- IF $R$ has no unity it is not ...
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1answer
23 views

I've proved everything about the ideal correspondence easily except $\pi ^{-1} \pi (\frak{a}) = \frak{a}$

The correspondence theorem to which I refer is the bijection between ideals of a commutative ring with $1$, $A$, and ideals of $A/\frak{b}$. I can prove easily most parts that imply the bijection ...
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1answer
36 views

Sum of Two Squares in Ring Theory

Show that a prime $p$ in $\mathbb{Z}$ is a sum of two squares iff -1 is a square in $\mathbb {Z}_{p}$. This example belong to my ring theory book didnt have ideal. i read in number theory that If ...
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Why is every non-zero element not a unit of this ring?

Consider the ring $\mathbb{Z}[\sqrt2]=a+b\sqrt2$ where $a,b\in\mathbb{Z}$. Now, if I am understanding the definition of units correctly, they are all the elements within the ring that have ...
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multilinear identity of degree 2

Let F be any field. A multilinear identity in $m$ indeterminates is an identity which has the form: $$f(x_1,x_2,\dots,x_m)=\sum_{\sigma\in S_n}a_{\sigma}x_{\sigma(1)}x_{\sigma(2)}\cdots ...
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1answer
47 views

Find all ring homomorphisms from $\Bbb Q$ to $\Bbb R$ [duplicate]

My question is find all homomorphism $ f: \Bbb Q \to \Bbb R$. I think I should use ring isomorphism theorem to do this problem, but I just don't know how to do this.
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1answer
41 views

In Ring Theory, does a 'power' of a morphism represent composition?

Say there is a ring homomorphism, denoted by $\theta$. If the notes use the expression $\theta^2$, then are they referring to the composition of the $\theta$ homomorphism with itself?
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25 views

Prime Factorization of 6

What would be the prime factorization of 6 in $Q[√−1]$? Can I generalize this to other numbers as well or no? Can someone please help me here?
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1answer
23 views

Quadratic Prime integer norm is not prime

What would be an example of a quadratic integer in $Q[√−1]$ which is prime, but whose norm is not prime?
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1answer
67 views

What does the notation $\mathbf{R}^\mathbf{R}$ mean?

I was reading the Princeton Review of GRE math subject test (4th edition), and one question was (page. 251) Example 6.24 Is the ring $\mathbf{R}^\mathbf{R}$ an integral domain? ...
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4answers
64 views

Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$

I was working on this problem Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$. My attempt:- I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should ...
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0answers
25 views

Let I be an unmixed radical ideal of R. then (I:x) is unmixed

Let $R$ be commutative ring with $1$. One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors (in other words,􀀀 if the associated prime ideals of $R/I$ are the minimal prime ...
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2answers
42 views

is 0 in the following Ideal?

Given $R=\mathbb R[x]$ and $I=(2x^3-3x^2+2x-3)+(2x^2-x-3)$ Is an Ideal of R? I don't understand what the quantity I is... Am I supposed to sum them together giving $2x^3-x^2+x-6$ Now here's the ...
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1answer
29 views

Quadratic number field which is Euclidean but not norm Euclidean

I am looking for an example of a quadratic field $\mathbb Q[\sqrt d]$ , with $d \equiv 2 $ or $3(\mod 4)$ , whose ring of integers is Euclidean but not norm (http://en.wikipedia.org/wiki/Field_norm ) ...
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votes
1answer
31 views

Finite field extension over the rationals need only one generator?

Some book stated (without proof) that in every finite dimensional field over the rationals of dimension $n$, there is an element of degree $n$ (i.e. any field $Q[\alpha_1, ..., \alpha_n]$ is of the ...
2
votes
2answers
32 views

Nilpotent elements in the quotient ring of a polynomial ring

If $F$ is a field and $p(x) \in F[x]$, prove that the ring $R=F[x]/(p(x))$ has no nonzero nilpotent elements iff $p(x)$ is not divisible by the square of any polynomial. (==>) $R$ has no ...
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1answer
36 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
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1answer
38 views

show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]. $ where $\Bbb F $ is a field

Let $\Bbb F $ be a field and $f(x)=\sum_0^n a_i x^i \in \Bbb F[x]$. Show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]$ Show that if $r \neq 0$ is a root of $f(x)$ then $r^{-1}$ is a root of $g(x)$ Find ...