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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2
votes
1answer
49 views

Is $(a,b)/(b)$ equal to $(a)/(b)$?

I'm doing ring theory, and I'm trying to understand quotients and ideals a little bit better. I was playing around a little bit with definitions. Can I say that this is true: $(a,b)/(b)$ = $(a)/(b)$ ...
2
votes
1answer
176 views

$f:\mathbb Z[x] \rightarrow\mathbb Z[x], f(x) = x^2$ is a ring homomorphism?

$f:\mathbb Z[x] \rightarrow\mathbb Z[x], f(x) = x^2$ is a ring homomorphism? Say I take two elements from $\mathbb{Z}[x]$. i.e. Say I take $a_0 + a_1 x + a_2 x^2 + ... + a_n x^n$ and $b_0 + ...
3
votes
1answer
95 views

Definition of Ring Homomorphism

I am using a text right now for abstract algebra ("A Concrete Introduction to Abstract Algebra" by Lindsay Childs) that seems to use a non-standard defn of ring homomorphism. I want to see if others ...
0
votes
0answers
27 views

Calculations with quotients in ring theory. Which rules are true?

Let $R$ a ring and let $(a),(b)$ principal ideals. Is it true that $$R/(a,b)=\frac{R/(a)}{(b)}?$$ I'm reading a book about rings, and it seems that they are using all kind of those tricks. But I'm ...
4
votes
2answers
165 views

Consider the ring $R=ℂ[X,Y]$ and the ideal $I=(X^2-Y,X^2+Y)$. We find (??) that $R/I ≅ℂ[X]/(X^2)$.

I'm trying to understand a step in an example of my reader about rings. Consider the ring $R=ℂ[X,Y]$ and the ideal $I=(X^2-Y,X^2+Y)$. We find that $R/I ≅ℂ[X]/(X^2)$. As the author doesn't ...
1
vote
2answers
67 views

Are these definitions of a prime ideal equivalent?

I just noticed I have three different definitions of a prime ideal in my notes. So are these definitions equivalent? Are they all correct...I have feeling I might have taken something down wrong. Let ...
10
votes
4answers
722 views

If a subring of a ring R has identity, Does R also have the identity?

I know it does not make sense that if a subring of a ring R is commutative, then R is also commutative. (For example, the set consisting of the matrices whose all entries except (1,1)-entry are zero, ...
0
votes
1answer
31 views

Show that $(p,X)/(pℤ[X])$ isomorph to $(X)$

Let $p$ prime. Let $(p,X)$ the ideal generated by $p$ and $X$ of the ring $ℤ[X]$. Show that $(p,X)/(pℤ[X])$ isomorph to $(X)$ where $(X)=X ℤ_p[X]$ I think I need to use that if $f:R → R'$ a ...
0
votes
2answers
82 views

Let $R$ a commutative ring and let $a\in R$. What does $aR$ mean?

Let $R$ a commutative ring and let $a\in R$. What does $aR$ mean ? I would think it means $\{ar : r \in R \}$ as that was the meaning in group theory. The thing that confuses me is that in group ...
1
vote
1answer
100 views

Which of the following statement is not necessarily true for the product of rings $R \times R$ when it is true for $R$?

$R$ is a ring. Which of the following statements is not necessarily true for the product of rings $R \times R$ when it is true for $R$? A. There exists some generator whose order is finite. B. $R$ ...
4
votes
5answers
1k views

The ring $ℤ/nℤ$ is a field if and only if $n$ is prime

Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$ is prime. Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any ...
2
votes
4answers
45 views

Ideals in $Z_{24}$

The ideals in $Z_{24}$ are $(\overline{0}), (\overline{12}), (\overline{8}), (\overline{6}), (\overline{4}), (\overline{3}), (\overline{2})$ and $Z_{24}$ itself. Now why isn't, say, ...
2
votes
1answer
58 views

Four term exact sequence for Artin algebras

Suppose $\Lambda$ is an Artin algebra, i.e. an algebra finitely generated as $R$-module for a commutative Artin ring $R$, $A$ is a finitely generated left $\Lambda$-module, and $X$ a finitely ...
2
votes
3answers
410 views

Boolean rings have characteristic $2$

Let $R$ be a ring such that $a^2=a$ for all $a\in R$. Show that $a+a=0$ for all $a\in R$. I don't really understand what to do here. The only way that this would be possible is if $a=0$. So $R$ ...
1
vote
2answers
172 views

Show if $\phi$ is a ring isomorphism of $\mathbb{Z}\to\mathbb{Z}$, then $\phi$ is the identity mapping.

Show if $\phi$ is a RING isomorphism of $\mathbb{Z}\to\mathbb{Z}$, then $\phi$ is the identity mapping. I don't really know where to start with this one. I know that since $\phi$ is an isomorphism, ...
3
votes
2answers
451 views

The ring of convergent power series over $\mathbb C$ isn't noetherian

How can one prove that the ring of convergent everywhere power series in $\mathbb C[[z]]$ isn't Noetherian?
0
votes
2answers
92 views

Nilradical of $\mathbb{R}[X,Y]/(X^nY^m)$

I'm having trouble finding the nilradical of $\mathbb{R}[X,Y]/(X^nY^m)$ for given $n$ and $m$. I believe the nilradical is $\{f(XY) \in \mathbb{R}[XY] : f \textrm{ has constant term 0}\}/(X^nY^m)\}$. ...
0
votes
2answers
243 views

Structure of maximal ideals of the quotient $\mathbb{C}[x,y,z]/ I$

I am trying to understand the general approach to the problems of the following type: Problem. a) Let $I\subset\mathbb{C}[x,y,z]$ be an ideal generated by $$\langle \ (x^2+y^2)^3+zx+3y^2z^3\ ,\ ...
1
vote
1answer
51 views

Simple Calculation on Local Rings.

Let $p$ be prime and $\mathbb{Z}_{(p)}$ be the local ring. I already know, that \begin{align} \mathbb{Z}_{(p)}/p\mathbb{Z}_{(p)} \cong \mathbb{Z}/p\mathbb{Z}. \end{align} What ist the explicit map? ...
2
votes
2answers
114 views

Endomorphisms of modules satisfying chain conditions and counterexamples.

1) How can I show that any one to one endomorphism of an Artinian module is an automorphism. 2) I also want to show that any onto endomorphism of a Noetherian module is an automorphism. I need ...
6
votes
2answers
294 views

Centre of a simple algebra is a field

How can one show that the centre of simple algebra is a field? I have tried it and proved that the inverse exists for every element of centre but cannot prove that inverse of every element ...
1
vote
1answer
62 views

How to show that a ring is semilocal?

Let $R$ be a commutative, local ring and let $f$ be a monic polynomial in $R[x]$. How can I show that $R[x]/(f)$ is semilocal, respectively artinian? Thank you for your help!
2
votes
4answers
544 views

Example of a commutative ring with identity with two ideals whose product is not equal to their intersection

I need a specific example of a commutative ring with identity, and two ideals in the ring whose product is not equal to their intersection. I know that for two such ideals I and J, IJ = I ∩ J if I + ...
1
vote
2answers
74 views

Question concerning finite rings

Let $R$ be a finite ring. Is it possible that $R$ has an element $a\in R$ such that $a$ is a left divisor of zero and $a$ is not right divisor of zero? Thanks.
0
votes
1answer
62 views

A question about about ideals of rings

In ring $\mathbb{Z}/2\mathbb{Z}$, which polynomial is in the ideal generated by $1+x^2$ and $1+x^3$ $\mathrm{A}. 1+ x^4 \\ \mathrm{B}. x^5+x+1 \\ \mathrm{C}. 1+x^6$ This type of questions confused ...
1
vote
2answers
51 views

Ideals in $F[X]$ are of the form $(f(x))$ where $f$ can be chosen to be monic. How?

I am reading a statement whereby it says that In $F[X]$, where $F$ is a field, any ideal is of the form $(f(x))$ where $f$ can be chosen to be monic. I don't get this part of the statement '$f$ can ...
3
votes
1answer
162 views

$P$ is a prime ideal, and $ R/P$ has no nilpotent elements. Then $R/P$ is a domain.

Let $P$ be a prime ideal. Suppose that $R/P$ has no nonzero nilpotent elements. Show that $R/P$ is a domain. What I did : WTS : $(a+P)(b+P)=ab+P=0+P$ implies $a+P=0+P$ or $b+P=0+P$. but it ...
1
vote
2answers
158 views

Skew Laurent Polynomial Ring.

Let $R$ be a ring and $R[x^{\pm 1}]$ the Laurent Polynomial Ring. $R[x^{\pm 1}]$ is a domain since $R$ is. How to show this? Let $R$ be a ring and $R[x^{\pm 1}]$ the Laurent Polynomial Ring. If ...
2
votes
1answer
97 views

How to show that an extension is integral?

Let $R$ be a commutative ring and $I\subset R[x]$ an ideal in $R[x]$ that contains a monic polynomial. I want to show that $R/(R\cap I)\rightarrow R[x]/I$ is an integral extension. This is the ...
0
votes
1answer
95 views

If a union of ideals is closed under addition and multiplication, then all ideals are not prime

Let $J_1,\dots,J_n$, $n\geq 2$, be ideals of $A$, where $A$ is a commutative ring with unit. Suppose $X$ is a subset of $A$ closed under addition and multiplication, and $J_1,\dots, J_n$ is a minimal ...
0
votes
1answer
67 views

Finitely Related Module is the direct sum of a Free Module and a Finitely Presented Module

Q: Prove that any finitely related module may be expressed as the direct sum of a finitely presented module and a free module. Hint: If M is generated by X = X' U X'', where X' is the finite ...
1
vote
1answer
36 views

A certain ideal of a valuation ring

This is a question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
0
votes
5answers
522 views

Is $x^8+1$ irreducible in $\mathbb{R}[x]$

Question is to check if : $x^8+1$ is irreducible over $\mathbb{R}[x]$. even before this I tried to see $x^4+1$ and $x^2+1$. for $x^2+1$, it does not have a root in $\mathbb{R}$ So, it is ...
0
votes
1answer
49 views

Existence of a natural linear isomorphism between certain group rings

Given a set $X$, let $J(X)$ denote the free monoid on $X$ and $F(X)$ denote the free group on $X$. Let $k$ be a field of prime characteristic. Since $F(X)$ and $J(X)$ have the same cardinality, there ...
0
votes
1answer
86 views

Decomposable Tensors over Rings

Suppose $R$ is a commutative ring and $M$ is a $R$-module. Then we can define the tensor product $M\otimes_R M$ and more generally the $k$-fold tensor powers $\otimes_R^kM$ for any $k\in\mathbb{N}$, ...
5
votes
1answer
865 views

Proof that $\mathbb Z[\sqrt{3}]$ is a Euclidean Domain

Let $R_d$ be the ring defined as $R_d=\left \{ x+y\omega : x,y\in \mathbb{Z} \right\}$, where $$\omega = \begin{cases} \sqrt{d}, & \text{if } \quad d \not \equiv 1\mod 4 \\ \frac{1+\sqrt{d}}{2}, ...
5
votes
3answers
144 views

When does there exist a commutative ring $C$ that contains rings $A$ and $B$ as a subring?

The statement I'm trying to prove is the following: Let $A$ and $B$ be commutative rings, both of characteristic $0$. Then there exists a commutative ring $C$ that contains both $A$ and $B$ as ...
3
votes
1answer
127 views

R is a UFD, c|ab, gcd(a,c)=1, then c|b

Let $R$ be a UFD and let $a,b,c \in R$. Prove that if $c|ab$ and $\gcd (a,c)=1$ then $c|b$. This is easy to prove if $R$ is a Euclidean domain, but I'm having trouble proving this for UFDs. I have a ...
4
votes
3answers
512 views

Prove : If $I = (p(x))$ is a prime ideal in $F[x]$ then $p(x)$ is irreducible.

I have to show : If $I = (p(x))$ is a prime ideal in $F[x]$, where F is a field, then $p(x)$ is irreducible. In the book I use, there is the proof of the converse which uses Euclid's Lemma. I ...
1
vote
1answer
127 views

Polynomial quotient ring : k[x,y,z,t]/(xy-zt)

I have some trouble picturing a quotient. Namely, what $k[x,y,z,t]/(xy-zt)$ looks like where $k$ is a field ? My intuition is probably wrong but is it isomorphic to $k[u,v,w]$ ?
2
votes
1answer
148 views

Example of “ring” without the distributive property?

Can anyone give an example of an "non-artificial" algebraic structure that fails to be a ring only because of a lack of one- and two-sided distributive property?
3
votes
2answers
123 views

Let $n \in \Bbb N$. Let $p>2$ a prime number. Show that $1^n+2^n+…+(p-1)^n \equiv 0 \pmod {p}$ [duplicate]

This is an exercise in my abstract algebra reader, in the chapter about polynomial rings. Let $n \in \Bbb N$. Let $p>2$ a prime number. And let $n$ not divisble by $p-1$. Show that ...
2
votes
1answer
119 views

Commutative ring addition where $a^2 = a$

I'm trying to solve following question: If $a^2=a$ for all $a \in R$ where $R$ is a commutative ring, then $a+a=0$. I have tried to solve this problem for a while now and I'm more or less stuck. I ...
1
vote
2answers
84 views

Injective modules under change of rings

Let $R$ be a ring with identity, $I$ an ideal and $M$ a left injective module with $IM= 0$. How can I show that $M$ is an injective $\frac RI$ module?
0
votes
1answer
145 views

Monomial ideals: isomorphism problem for commutative algebras?

Let $I,J\unlhd K[x_1,\ldots,x_n]=K[x]$ be monomial ideals and $f\!: K[x]\to K[x]$ a graded isomorphism (given by a matrix $A=[\alpha_{i,j}]\in K^{n\times n}$, i.e. $x_i\mapsto\sum_j\alpha_{i,j}x_j$ is ...
0
votes
2answers
47 views

which statement is correct?

$S$ is the power set of $\Bbb Z$. Define two binary operations: $+$ (the symmetric difference set which means $A+B=(A\cup B)\setminus(A\cap B)$ and $*$ (the intersection of two sets $A*B=A\cap B$, ...
0
votes
1answer
128 views

Finite dimensionality and maximal ideals

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Is the following equivalence true? A is finite-dimensional over $k$ if and only if $A$ has ...
5
votes
3answers
398 views

$1^n +2^n + \cdots +(p-1)^n \mod p =$?

Calculate for every positive integer $n$ and for every prime $p$ the expression $$1^n +2^n + \cdots +(p-1)^n \mod p$$ I need your help for this. I don't know what to do, but I'll show you what I ...
5
votes
1answer
79 views

$M$ is a free module iif $M\otimes B$ is free

Let $M$ be a module over some ring $A$ and let $B$ be some ring containing $A$ (or more generally let $\rho : A \to B$ be a ring homomorphism). Then we can endow $M \otimes_A B$ with a $B$-module ...
1
vote
1answer
83 views

Integral domain is ufd iff atomic and gcd domain

an Integral domain D is a UFD if and only if D is atomic and gcd domain. I know that a UFD is Atomic, but i don't know how to prove it is a gcd domain. the other side of the proof: I do understand ...