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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In the theorem is it necessary for ring $R$ to be commutative?

According to the statement of theorem that a commutative ring $R$ with prime characteristic $p$ satisfies $$\begin{align} (a+b)^{p^n} = a^{p^n} + b^{p^n} \end{align}$$ $$\begin{align} (a-b)^{p^n} = ...
0
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2answers
93 views

commutative ring with no zero divisors

Question from chapter on 'Integral Domains' from Gallian .which said an integral domain does not contains zero-divisor and is a commutative ring. Does there exist any other commutative ring which is ...
7
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3answers
2k views

“In a finite commutative ring, prove every element is a unit or zero divisor.” What happens if we drop “finite”?

This was my ring theory exam question which states: 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 ...
0
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1answer
102 views

Prove that a subset is a finitely generated subring

Consider $\mathbb{A}^2$ with $\rho : (x, y) \mapsto (-x, -y)$. Can anyone help me prove that $S = \{f \in \mathbb{C}[x, y] : f \circ \rho = f\}$ is a finitely generated subring? Also, can $S$ be ...
-1
votes
1answer
95 views

If $x,y$ are nilpotent and commute, $x+y $ is nilpotent. [closed]

Let $A$ a ring. Supose that $x,y \in A$ are nilpotents elements and that $xy=yx$. Prove that $x+y$ is nilpotent.
1
vote
1answer
132 views

$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : ring of integers, integral basis and discriminant

In the following document, http://people.math.carleton.ca/~williams/papers/pdf/033.pdf, I found three results about biquadratic fields and their ring of integers. It's the proof of the first theorem ...
0
votes
3answers
82 views

Prove that T is not a zerodivisor in A[T]

Let $A$ be any commutative ring. Consider the polynomial ring $A[T]$. Prove that $T$ is not a zerodivisor in $A[T]$. Generalize the argument to prove that a monic polynomial $$ ...
3
votes
1answer
687 views

Cancellation laws in Rings

In rings left and right cancellation laws generally don't hold. can anyone generalize some cases so that we are ensured when the cancellation laws hold in rings?(the case I found was in Integral ...
1
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2answers
52 views

Prove that $\mathbb{Z}[\zeta_{p} + \zeta_{p}^{-1}]$ is the ring of integers of $\mathbb{Q}(\zeta_{p} + \zeta_{p}^{-1})$

I'm a bit at a loss about what I can say in this situation. Do I have to show that $\zeta_{p} + \zeta_{p}^{-1}$ form an integral basis ? If I do, I have no idea how to do it. If not, can I use the ...
1
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3answers
164 views

Find the number of prime ideals (CSIR 2014)

Let $p,q$ be distinct primes. Then (1) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct ideals. (2) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct prime ideals. (3) $\dfrac{\mathbb{Z}}{p^2q}$ ...
2
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2answers
52 views

Showing that the $\text{ord}(a+b) = \text{min}(\text{ord}(a),\text{ord}(b))$ in a DVR

This is Problem 2.29 from Fulton's Algebraic Curves. First a bit of background because I don't know how standard his terminology is. For a discrete valuation ring $R$ with maximal ideal ...
3
votes
1answer
76 views

If $f_1,…,f_{n+1}\in\mathbb{C}[x_1,…,x_n]$, is there a polynomial in the coefficients which vanishes iff the $f_i$ have a common root?

My question is as in the title: Suppose $f_1,...,f_{n+1}\in \mathbb{C}[x_1,....,x_n]$. Is there polynomial $g$ (or a system of polynomials) with variables given by the coefficients of the $f_i$ ...
1
vote
2answers
106 views

Which of the following is also an ideal?

If $U,V$ are ideals of a ring $R$, then which of the following is also an ideal of $R$? $U+V=\{u+v\mid u\in U,v\in V\}$ $U\cdot V=\{u\cdot v\mid u\in U,v\in V\}$ $U\cap V$ My attempt: I have ...
2
votes
2answers
41 views

$\operatorname{rad}(I)=\bigcap_{I\subset P,~P\text{ prime}}P$

$R$ commutative ring with unity. $I$ $R$-ideal. Then $\operatorname{rad}(I)=\bigcap_{I\subset P,~P\text{ prime}}P$. That is, the radical of $I$ is the intersection of all prime ideals containing $I$. ...
4
votes
2answers
83 views

If $P \in \operatorname{Ass}M$, then $R/P \subset M$.

Let $R$ be a commutative ring with unity. $M$ an $R$-module. Then $P \in \operatorname{Ass}M$ if and only if there is a submodule $N\subset M$ such that $R/P \cong N$. ...
1
vote
1answer
52 views

Completion of absolute value on an integral domain

In the Wikipedia article on Absolute value (algebra), the completion of an integral domain is defined as the quotient ring of Cauchy sequences by null sequences. The integral domain is then embedded ...
0
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1answer
39 views

M is noetherian implies…

Im totally lost at just one sentence of a proof here. Let $$0 \rightarrow L \xrightarrow{\alpha} M \xrightarrow{\beta} N \rightarrow 0$$ be a s.e.s (it is exact) of $A$-modules (A is a commutative ...
0
votes
2answers
125 views

Checking whether a map satisfies being homomorphism

My question might seem silly ,but excuse me for it as I've just started my hand on Ring theory. My question is : Like we've in case of group homomorphisms that we can check that a map $\phi :G ...
1
vote
1answer
51 views

To Prove That Field of Fractions of Given Rings is Same. Proof Verification.

I am trying to solve Q. 8a in Section 9.1 from Abstract Algebra by Dummit & Foote. The problem is: Let $F$ be a field and $R=F[x,x^2y,x^3y^2,...,x^ny^{n-1},...]$ be a subring of $F[x,y]$. ...
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2answers
69 views

A question on endomorphisms of an injective module

This is a homework question I am to solve from TY Lam's book Lectures on Modules and Rings, Section 3, exercise 23. Let $I$ be an injective right $R$-module where $R$ is some ring. Let $H= ...
3
votes
1answer
71 views

Is it possible to extend a commutative ring to have a unity? [duplicate]

Let $R$ be a commutative ring. Then, is it possible to extend this to have a unity? That is, is there a commutative ring with unity $R'$ such that $R$ is a subring of $R'$?
0
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1answer
57 views

Any artinian chain ring is self-injective.

Let $R$ be an artinian chain (uniserial) ring. I want to prove that $R$ is self-injective, that is, any $\alpha:I\rightarrow R$ right $R$-module homomorphism can be extended to a homomorphism ...
1
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1answer
65 views

what does 2 ideals are equal mean?

I'm revisiting the proof of 1-1 correspondence theorem and while proving $f$ is one-one I don't know how to write mathematically what we mean by 2 ideals are equal? (Here $f$ is a map from set of ...
2
votes
1answer
64 views

Show the module of continuous functions of antiperiod $\pi$ is not cyclic.

Homework Hint Needed: Let $R$ be the ring of all continuous functions on $\mathbb{R}$ of period $\pi$. Let $S$ be the $R$-module of all continuous functions on $\mathbb{R}$ of antiperiod $\pi$. So ...
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vote
1answer
120 views

1-1 correspondence theorem

Here is the correspondence theorem stated as follows: Let $A$ be an Ideal of ring $R$.There is 1-1 correspondence between Ideals of $B$ containing $A$ and ideals of $R/A$. I have read the proof but ...
0
votes
1answer
29 views

The definition of $A'[\phi]$

I'm having trouble understanding the following definition from my textbook: If $M$ is an $A$-module and $\phi: M \longrightarrow M$ an $A$-linear endomorphism of $M$, I write $A'[\phi] \subset$ ...
2
votes
2answers
104 views

Is the right annihilator of an element in a ring a subring?

let $R$ be a ring and let $x$ not equal to $0$ be a fixed element in R. Then is $\{r \mid xr=0\}$ a subring of $R$? The solution says yes, but I don't think so, because the multiplicative identity is ...
2
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2answers
241 views

Ring with convolution product

Let $M$ be a monoid and $R$ a ring, $f,g \in R^{(M)}$ (the functions from $M$ to $R$ with finite support), we define the convolution product as $$x\in M \implies (f*g)(x)=\sum_{yz=x} f(y)g(z).$$ Show ...
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1answer
33 views

Ideals in a Ring

I completed part a with no problems using the ideal test. However it is part b that is giving me troubles. I'm not sure where to start with this one, any help is much appreciated!!!
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2answers
57 views

Sum of the elements of a cyclic subgroup

Let $G$ be a finite cyclic subgroup of the group of units of a commutative unital ring $R$. What is the sum of the elements of $G$, i.e. $$\sum_{x \in G}{x}?$$ [The answer is not difficult in the ...
2
votes
1answer
33 views

Ring Theory: Ideals

Let $S$ be a nonempty subset of $R$. Let $r(S)=\{x \in R|Sx=0\}$ and $l(S)=\{x \in R|xS=0\}.$ Show that $r(S)$ and $l(S)$ are ideals in $R$ if $S$ is an ideal in $R.$ Here is my attempted solution ...
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1answer
25 views

Writing a direct product of rings over a ring “as one”.

Let $R$ be a commutative ring and suppose that $R/I \oplus R/J$ as ring where I and J are ideals not coprime. Suppose that I wanted to write this as $R[x_1,\ldots ,x_n]/K$ where K is an ideal of the ...
0
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1answer
69 views

Relation between ideals in Noetherian domains.

Suppose that we have a Noetherian domain $R$ and two ideals $I$ and $J$ of $R.$ Now consider the minimal (or irredundant) primary decompositions $I=\bigcap\limits_{i=1}^r Q_i$ and ...
0
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1answer
55 views

Ring morphisms, group ring problem

Prove that if $A$ is a ring and $G$ is a group, then the map $$Hom(\mathbb Z[G],A) \to Hom(G,\mathcal U(A))$$ which sends $f \rightarrow f|G$ is a bijection. First of all, I am having some problem ...
1
vote
1answer
99 views

Jacobson radical of a matrix ring

I search for a way to prove that the Jacobson radical of $R=\left [\begin{array}\ \mathbb Z_4 & 2\mathbb Z_4 \\ 0 & \mathbb Z_4 \end{array} \right ]$ is $\left [\begin{array}\ 2\mathbb Z_4 ...
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2answers
118 views

Intersection of prime ideals in a chain is prime

I'm trying to prove that the intersection of prime ideals in a chain is again prime. It seems easy enough for any pair of prime ideals: let $P$ and $P'$ be two prime ideals in the chain. Assume ...
3
votes
2answers
125 views

On modules over simple rings

We suppose that all rings are left Artinian simple rings and all modules over a ring are of finite length. Let $M \neq 0$ be a left module over a ring $R$. By Wedderburn theorem, $R$ is a matrix ring ...
3
votes
2answers
136 views

Showing an ideal with maximality condition is prime.

Let $R$ be a commutative domain and suppose that $I \subseteq R$ is an ideal of $R$ maximal with respect to the property that $I^{-1} \not\subseteq R$. Show that $I$ is a prime ideal. This is ...
0
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1answer
52 views

In the integral domain $D = \{r+s \sqrt{17}: r,s \in\Bbb Z\}$, which element is irreducible?

In the integral domain $D = \{r + s \sqrt{17} | r,s \in\Bbb Z\}$, which is irreducible? $3 - \sqrt{17}$ $9 - 2\sqrt{17}$ $7 + \sqrt{17}$ $13 + \sqrt{17}$ I got all of them are irreducible, if you ...
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1answer
49 views

Proving $(aR)(bR)=(abR)$ in a commutative ring

To do this proof I'd like to get the conclusion from the title from an easier statement: $(a)(b)=(ab)$ for $a,b\in R$. If I'm right, in a commutative ring the principal ideal $(a)$ is generated by ...
2
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1answer
89 views

Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings.

I'm trying to do a few proofs about rings: $(1)$ I want to show that for a ring $R$ and the ring of matrices $M_n(R)$, if $I\lhd R\implies M_n(I)\lhd M_n(R)$ and that all ideals of $M_n(R)$ are of the ...
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1answer
59 views

For any two Ideals $A$ and $B$,$A+B=\langle A \cup B \rangle$

Below is the proof of : Prove that for any two ideals $A$ and $B$ of ring $R$,$A+B=\langle A \cup B~\rangle$ . Proof: By theorem (for any two ideals of a ring $R$ ,then the set $A+B$ is an ...
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1answer
438 views

generators of ring

I've just started with ring theory and want to understand what in actual does it mean by generators of a ring. As in group theory we have the set of generators as those elements on which if we ...
3
votes
1answer
477 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...
15
votes
1answer
427 views

Rings with $a^5=a$ are commutative

Let $R$ be a ring such that $a^5=a$ for all $a \in R$. Then it follows that $R$ is commutative. This is part of a more general well-known theorem by Jacobson for arbitrary exponents ($a^n=a$), which ...
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0answers
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Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
3
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2answers
146 views

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the ...
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3answers
80 views

Ideal generated by a subset of ring.

The definition of Ideals generated by a subset : Let $S$ be any subset of ring $R$ then an ideal $I$ of $R$ is said to be generated by $S$ if : (1) $S \subseteq I$. (2) for any ideal $J$ of $R$ ...
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2answers
27 views

The subring criterion

As we know that the subring criterion states that a subset $H$ of ring $R$ is a subring if and only if : (1) $H$ is non-void , and (2) for all $x,y \in H$,$x-y \in H$. (3) product $xy \in H$ . The ...
2
votes
1answer
69 views

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite.

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite. 2) Classify $(\Bbb Z[\sqrt2]^*, .)$, where $\Bbb Z[\sqrt2]^*$ is the group of units of $\Bbb Z[\sqrt2]$ What I have done so far that for $a+b\sqrt2$ ...