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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Laurent Polynomials Ring

Wikipedia says: "The Laurent polynomial ring $R[X, X^{−1}]$ is isomorphic to the group ring of the group $\mathbb{Z}$ of integers over $R$". Can anyone offer a proof? I also don't fully understand ...
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Basis of Vector space $\Bbb C$ over rational numbers.

What will be the basis of vector space $\Bbb C$ over field of rational numbers? I think it will be an infinite basis! I think it will be $B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$. ...
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Prove-If n is prime $\mathbb{Z}_n$ is a field.

I need to prove that $\mathbb{Z}_n$ is a field if and only if $n$ is prime. And I proved the forward. But I am not sure how to prove the backward, 'if n is prime $\mathbb{Z}_n$ is a field. ' What ...
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1answer
88 views

Embedding of continuous functions into differentiable functions

This question refers to a solution printed in the current (December 2013, 120(10)) issue of The American Mathematical Monthly, p. 944. There, the authors intend to show that any ring homomorphism ...
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42 views

Question about zero-divisors , rings and polynomials.

Let $i,n,m$ be positive integers. For every nonnegative integer $k<i+1$ , let $a_k$ be elements of a ring $A$ that satisfies : $1)$ The ring $A$ is isomorphic to ${\Bbb R}^{n}\times{\Bbb ...
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86 views

$p$-adic completion of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$.

Let $p$ be prime. The $p$-adic completion of $\mathbb{Z}$ is the ring $\mathbb{Z}_p$ of $p$-adic integers, and its elements can be thought of as power series in $p$. Is there a nice description of the ...
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1answer
65 views

Are two different prime ideals relatively prime?

Are two different prime ideals relatively prime? Thanks in advance!
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137 views

Subring of the field of rational numbers

Let $R=\{a\cdot2^n\mid a,n \in \mathbb{Z}\}$ be a subring or the field of rational numbers $\mathbb Q$. i) What kind of elements are invertible in $R$? ii) Prove that $R$ is a principal ...
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Examples of Dedekind rings with infinite class number

I am looking for explicit examples of Dedekind rings with infinite class number. In most books on algebraic number theory there is a standard example (before or after proving that the class number is ...
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2answers
64 views

Comparison between rings and groups Question

Show by example, that for nonzero (fixed) elements a & b in a ring, the equation ax=b can have more than one solution. How does this compare to groups? Can someone help me compare rings to ...
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1answer
30 views

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$.

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. It suffices to prove for the case $F=\mathbb{Z}_p$. How to prove?
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Is there a finite commutative semigroup $S$ with $S^2 = S$ which is not a monoid?

Is there a finite commutative semigroup $S$ with $S^2 = S$ which is not a monoid? where $S^2=\{ab\mid a,b\in S\}$. As is known, if such $S$ can be a ring with an addition then it is a monoid? So if ...
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145 views

Binary Representation of Complex Numbers

Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA has finite models based on modular arithmetic. MA ...
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3answers
234 views

similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
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1answer
149 views

isomorphism between factor ring of matrices and Z

I have a commutative ring R= $\begin{pmatrix}a & b \\ 0 & a \end{pmatrix}$ (R is a 2x2 matrix, a, b $\in$ Z), I=$\begin{pmatrix}0 & b \\ 0 & 0 \end{pmatrix}$ is an ideal. I need to ...
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1answer
26 views

Unity Question dealing with Rings [duplicate]

The ring $\{0, 2 ,4, 6, 8\}$ under addition and multiplication modulo $10$ has a unity. Find it. Is the unity because this is only a set of evens? I don't really understand unity. Can someone help?
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1answer
117 views

Discriminant is zero iff $f\in K[X]$ has repeated roots

I have to prove the statement in the title. Proving from right to left is easy. Now from left to right: $D=(\alpha_1-\alpha_2)^2(\alpha_1-\alpha_3)^2\cdots(\alpha_{n-1}-\alpha_n)^2$ where $\alpha_i$ ...
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1answer
79 views

What ring is the quotient $\mathbb{Z}[\sqrt{-11}]/(3,1+\sqrt{-11})$ isomorphic to?

Could anyone help me with this question? I've the feeling that the answer is $\mathbb{Z}/3\mathbb{Z}$, but I'm not sure at all and above all I don't know how to prove it. Thanks
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1answer
243 views

Find unity of ring.

The ring {0, 2, 4, 6, 8} under addition and multiplication modulo 10 has a unity. What is that unity and how do we find it?
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2answers
50 views

Abstract Algebra: Struggles with rings

This is a part of the group of practice problem I've been working on and I'm just lost. I'm really struggling when it comes to these ring problems. Anybody who could lay out an outline for this ...
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1answer
68 views

Abstract Algebra: Rings and Units.

Let $u$ be a non-zero element of a simple, commutative, unital ring $R$. Show that $u$ is a unit of the monoid $(R, *, 1)$. I'm just really struggling to understand the concept of building or ...
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360 views

Give an example of a finite non-commutative ring.

Similarly, give an example of an infinite non-commutative ring that does not have a unity.
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175 views

On the center of a ring

Let $R$ be a ring. The center of $R$ is the set $\{x\in \!\,R\mid ax=xa \text{ for all }a \in R\}$. Prove that the center of a ring is a sub-ring. I am not sure how to start off this problem.
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1answer
238 views

Let $I$ be an ideal of a ring $R$, and let $S$ be a subring of $R$. Prove that $I\cap S$ is an ideal of $S$

Let $I$ be an ideal of a ring $R$, and let $S$ be a subring of $R$. Prove that $I\cap S$ is an ideal of $S$ I've tried listing the properties of an ideal, but I don't know where to go from there.
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Multiplicative structure on additive group

Let $R$ be a ring without assumption of existence of unity. Let $R^{\ast} = R \oplus \mathbb{Z}$ as abelian groups. Show how to define multiplication on $R^\ast$ so that it becomes a ring with an ...
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1answer
195 views

Prove a commutative ring with characteristic n has a subring isomorphic to $\mathbb{Z}_n$

Let $R$ be a commutative ring with identity such that the characteristic of $R$ is $n$, char$R=n$. Prove that is $n>0$ then $R$ contains a subring isomorphic to $\mathbb{Z}$$_n$, the additive ...
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792 views

Finite rings without zero divisors are division rings.

How can I prove this: Finite rings without zero divisors are division rings. I know how to prove it, when I add, that my ring has 1, but i have no idea, if my ring needs to have an unity.
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1answer
45 views

How to correctly write this ring theoretic thing?

Im unsure how to write this thing below in a formal way : For an integer $n>2$ Let $F_n(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}.$ Also we have $x^n = 1$ and $1 + x + x^2 + ... + ...
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566 views

Maximal ideals in the ring of Gaussian integers

Let $R= \{ a+bi : a,b \in \mathbb{Z} \}$ be a subring of $\mathbb{C}$. Consider two principal ideals $I=(7)$ and $J=(13)$ in $R$. Is the ideal $I$ maximal? How about $J$? I don't understand what ...
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80 views

How is this map a well-defined homomorphism?

If $f: R \rightarrow S$ is a homomorphism of rings with kernel $K$, and $I$ is an ideal in $R$ such that $I \subset K$. The hypothesis is that the map $\overline{f}: R/I \rightarrow S$ given by ...
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1answer
79 views

Let $P_0 \subsetneq P_{1} \subsetneq \dotsb \subsetneq P_n$ be a chain of prime ideals in a Noetherian ring.

Let $P_0 \subset P_{1} \subset \dotsb \subset P_n$ be a chain of prime ideals in a Noetherian ring. Show that if $x \in P_n$ then there exists a chain of prime ideals $Q_{1} \subsetneq \dotsb ...
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1answer
56 views

Why is $K(\alpha) = \left\lbrace \frac{f(\alpha)}{g(\alpha)} : f,g\in K[X],\, g(\alpha)\neq 0\right\rbrace$?

I know from definition that: $K(\alpha)$ denotes the smallest subfield of $L$ that contains both $K$ and $\alpha$. I've read here that this is equivalent with: $$K(\alpha) = \left\lbrace ...
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Ring theory : Completely lost and overwhelmed

Over the past 3( 9 sessions) weeks my professor has covered entire Part 3 - Rings from Gallian's Abstract Algebra which includes Introduction to Rings Motivation and Definition Examples of ...
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1answer
50 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
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282 views

When are kernels of homomorphisms on $\mathbb{Z}[x]$ principal ideals, and what does this have to do with Gauss's lemma?

Two questions listed in Artin under a section called "Gauss's lemma": Are the kernels of $f(x) \mapsto f(1 + \sqrt{2})$ and $f(x) \mapsto f(\frac12 + \sqrt2)$ (for $f \in \mathbb{Z}[x]$) principal ...
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111 views

Using formal power series.

Suppose $R$ is a commutative ring (with or without $1$), and for $a_0,...,a_m,b_0,...,b_m\in R$ and for all $0\le k\le m$ we have $$a_0b_k+a_1b_{k-1}+\dots+a_kb_0=0 $$ then there's some nonzero $r\in ...
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1answer
188 views

Krull's Height Theorem Proof.

http://www2.gsu.edu/~matfxe/commalglectures/lect14.pdf Here is a proof of Krull's PIT theorem. I don't understand why $\cap_{t\geq 1} (PR_{P})^t = 0$ (written on Page 2, line 3) If anyone would shed ...
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2answers
847 views

How to prove the ring of Laurent polynomials over a field is a principal ideal domain?

Disclaimer: this is a homework question. I'm looking for direction, not an answer. Given a field $F$, show that $F[x,x^{-1}]$ is a principal ideal domain. I'm unsure how to proceed. Would it be ...
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2answers
142 views

What is the difference between a division ring and a quotient ring?

What is the difference between a division ring and a quotient ring ? Im confused. Can a quotiënt ring be defined without an ideal ? Can a division ring be defined with an ideal ? I guess some division ...
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1answer
142 views

R ring is noetherian, commutative, unitary and integral domain, is R a field?

This is the question: "let R be a commutative unitary ring that is also integral domain and noetherian, prove that R is a field" I'm having some trouble proving this. For R to be noehterian means ...
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1answer
56 views

finding the isomorphism between rings

How can I find the isomorphism between rings: $ \Bbb Z [T] / (T^2-5T+6) $ and $ \Bbb Z^2 $? $ \Bbb Z [T] / (T^2-5T+6) = \Bbb Z [T] / (T-3)(T-2) $. Is is true, that: $\Bbb Z [T] / (T^2-5T+6) = \Bbb Z ...
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Conjecture about some rings and roots of unity. [duplicate]

Let $\Bbb R_{\geqslant 0}[X_n]$ be a polynomial semiring. More precisely $\Bbb R_{\geqslant 0}[X_n]$ are the polynomials of $X_n$ with positive real coefficients with $(X_n)^n = 1$. Let $F(n)$ be ...
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58 views

elementary symmetric polynomial

How to make an elementary symmetric polynomial from $[ \ (X-Y)(X-Z)(Y-Z) \ ]^2$ all I can do is this: $ [ \ (X-Y)(X-Z)(Y-Z) \ ]^2 = [ \ X^2Y -X^2Z -XYZ +XZ^2 -XY^2 +XYZ +Y^2Z -YZ^2\ ]^2 = [\ X^2Y - ...
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1answer
29 views

Simple Ring Question

Let $R$ be a commutative ring with identity. Why does $(-1)(a) = -a$? This should be true if and only if $a + (-1)(a) = 0$.
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60 views

Irreducible polynomial in $\mathbb{C} [x,y ]$

I want to prove that $t^4+xt^3+yt^2+xt+1$ is irreducible over $\mathbb{C}[x,y]$. I know that $\mathbb{C}[x,y][t]$ is a UFD just as $\mathbb{C}[x,y]$ because the base ring is a field. Can I just ...
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1answer
44 views

What does it mean to localise a ring, at a prime, in this context?

Given a chain of prime ideals of a commutative ring $R$, $P_1 \subseteq P_2 \subseteq Q$. Then what does it mean to localise the chain at $Q$? I know what the localisation of a ring is. But I don't ...
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117 views

Is $(x)$ a maximal ideal in $\mathbb{Z}[x]$?

It seems to me that it is as I can't think of another ideal that will 'contain' it apart from $\mathbb{Z}[x]$ itself?
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1answer
62 views

Prove $f(x)=9x^2-5y^2-34$ has no integral roots

Prove $f(x)=9x^2-5y^2-34$ has no integral roots. I have tried working this mod 2, 3, 4, 5, and 17, and some random others, to no avail. It is for a graduate course, so I am thinking maybe I tried to ...
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4answers
148 views

Commutative ring and zerodivisor

Let $a \ne 0$ belong to a commutative ring. Prove that $a$ is a zero divisor if and only if $a^2b=0$ for some $b \ne 0$. I know that to be a zero divisor there has to be a non zero element $b$ ...
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1answer
43 views

Determining whether $2x ^ 3 - x^2 + 1$ irreducible in $\mathbb{Q}[x]$?

I'm checking if $f(x) = 2x ^ 3 - x^2 + 1$ is irreducible in $\mathbb{Q}[x]$. We have a polynomial of degree three so if it has no roots it is irreducible. But as we are dealing with $\mathbb{Q}[x]$ ...