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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image of the map of rational numbers

Given a map $\phi: {a\over b} \rightarrow \bar{a} \cdot (\bar{b})^{-1} \pmod p$, where $a,b\in\mathbb Z$ and $p$ does not divide $b$ show that $\operatorname{Im}(\phi)= \mathbb Z_p$. Anyone can help? ...
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2answers
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Non boolean example of a finite ring $R$ with $r^4 = r$ for all $r$ in $R$.

I just proved that a finite ring $R$ with $r^4 = r$ for all $r$ in $R$ must be commutative. But I don't see any non boolean example to ilustrate.
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References in Ring Theory

I am very interested in ring theory. I want to know the best textbooks about that subject, i.e about PID's, UFD's, Dedekind domain, Euclid domain. Can anyone tell me?
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Why $\langle I, J\rangle =R$ for distinct prime ideals $I$, $J$ of a principal ideal domain $R$?

Let $R$ be a principal ideal domain with identity and $I$, $J$ be distinct prime ideals of $R$. Prove that $1 \in \langle I, J\rangle$ hence $\langle I, J\rangle = R$. How to prove?
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Halving One in Odd Size Rings

Consider the rings $\mathbb{Z} /n \mathbb{Z}$ where $n$ is odd. Every number is even in such rings. Assume we start with $1$ and keep "halving" until we get back to $1$. What can be said about the ...
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1answer
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Let $M$ be a maximal ideal of a ring $R$. Is $M[x]$ a maximal ideal of $R[x]$?

Let $M$ be a maximal ideal of a ring $R$. Prove that $M[x]$ is not a maximal ideal of $R[x]$ Thanks a lot
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1answer
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Prove that $R\left[x\right]/I\left[x\right]\cong\left(R/I\right)\left[x\right]$ [duplicate]

Let $I$ be an ideal of a ring $R$, define $I[x]$ to be the set of all polynomials whose coefficients are in $I$. Prove that $R\left[x\right]/I\left[x\right]\cong\left(R/I\right)\left[x\right]$ Help ...
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Prove that every non-zero element of a finite commutative ring with unity is either a zero-divisor or a unit. [duplicate]

Let R be a finite commutative ring with unity. Prove that every non-zero element of R is either a zero-divisor or a unit. What happens if we drop the "finite" condition on R ?
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1answer
53 views

Elements of Group Rings, Are they Reducible?

In general, elements of group rings are written as sums of the group elements multiplied by scalars from the ring, correct? What is the utility of that, if we don't know how to reduce the elements of ...
3
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1answer
98 views

Units in finite polynomial rings

Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where ...
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657 views

Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal

Let $R$ be a ring. Prove that each element of $R$ is either a unit or a nilpotent element iff the ring $R$ has a unique prime ideal. Help me some hints.
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1answer
556 views

Prime/maximal ideals of $\mathbb{C}[x, y]$ containing a given ideal

Remember that (i) every maximal ideal is a prime ideal, (ii) for proper ideals $I$ of rings $R$, the factor ring $R/I$ is a field iff $I$ is a maximal ideal of $R$, and that (iii) whenever $F$ (for ...
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0answers
53 views

Question about finite ring more than one element; division ring [duplicate]

A finite ring with more than one element and no zero divisors is a division ring. This a problem taken from Hungerford's graduate algebra text. Hungerford defines left and right zero divisors and ...
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2answers
436 views

Kernel of a map $\mathbb{C}[x,y,z] \to \mathbb{C}[t]$ [duplicate]

I'm having issues with a question in Artin, more specifically 11.3.3.e. The question asks: Find generators for the kernel of the following map: $\mathbb{C}[x,y,z] \to \mathbb{C}[t]$ given by $x ...
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1answer
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Question about Units

Find a nonzero element in a ring that is neither a zero-divisor nor a unit. Would Z7 work for this example? I know it has no zero divisors, but I am confused about the unit portion. Can someone ...
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2answers
109 views

Ring of polynomials $\mathbb{Z}/(n)[x]$

I'm doing problems in Artin on rings, and in problem 11.2.1, he asks: For which positive integers n does the polynomial $x^2+x+1$ divide $x^4+3x^3+x^2+7x+5$ in the ring $\mathbb{Z}/(n)[x]$? ...
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207 views

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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3answers
296 views

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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3answers
64 views

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
87 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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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 ${\Bbb R}^{n}\times{\Bbb ...
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$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
63 views

Are two different prime ideals relatively prime?

Are two different prime ideals relatively prime? Thanks in advance!
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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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1answer
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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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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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0answers
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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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225 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
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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
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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
115 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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3answers
702 views

Show, by example, that for fixed non-zero elements a and b in a ring, the equation ax = b can have more than one solution. [closed]

Show, by example, that for fixed non-zero elements a and b in a ring, the equation ax = b can have more than one solution. How does this compare with groups?
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1answer
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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
237 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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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
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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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350 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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168 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
231 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
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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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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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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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2answers
535 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
55 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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2answers
308 views

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 ...