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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3
votes
2answers
111 views

A proof and question about Quotient ring

I don't understand quotient rings very well, and I am confused about the proof of "The quotient ring $\Bbb Z/(m)$ is a field if and only if $m$ is a prime." I know what mod means. Help me understand ...
0
votes
1answer
103 views

questions related to Hilbert basis theorem

Let $A$ be a commutative ring with unit. How to do the following questions related to Hilbert Basis Theorem? I am quite confused about the proof of Hilbert Basis Theorem. If $A[x]$ is Noetherian ...
2
votes
1answer
61 views

Counterexample of certain non-primary ideals

Let $A$ be a commutative ring with unit. Let $I,J$ be primary ideals of $A$ such that $J$ is not contained in $I$ and $r(J)\subset r(I)$, $r(J)\neq r(I)$. Then $I\cap J$ is not necessarily primary. ...
0
votes
0answers
68 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...
2
votes
1answer
35 views

Proving that $\{f \in End(A): \forall a \in A:|a|<\infty \implies f(a)=0\}$ is an ideal

I need to prove that $I = \{f \in End(A): f(a)= 0 \ \text{for all $a \in A$ with a finite order}\}$ It isn't hard to prove that $I$ is a subgroup of $End(A)$, but it is quite hard to prove that: ...
0
votes
3answers
78 views

What is $ℂ[X]/(X^2+1)$ and why is this not a maxmial ideal.

This is a passage in a book I'm reading about ring theory: The ideal $(X^2+1)$ is maximal in $ℝ[X]$. In $ℂ[X]$ this ideal is not maximal. I understand that $ℝ[X]/(X^2+1)=ℂ$ therefore it is a ...
1
vote
3answers
86 views

Interpreting the set $IJ = \{\sum_i x_iy_i \mid x_i \in I, y_i \in J\}$ where $I$ and $J$ are ideals

Let $I$ and $J$ be ideals in a ring $R$. Show that $IJ = \left\{\sum_i x_iy_i \mid x_i \in I, y_i \in J\right\}$ is an ideal. Question I am not sure how to interpret this question because of the ...
1
vote
2answers
128 views

Generating set for a polynomial ideal

I would like to know which is the generator set for the following polynomial ideal: $$ I=\{a_nx^n+\cdots +a_0\in\mathbb{Z}[x]\,\, | \,\, a_0\,\, \text{is even}\}. $$ Sorry for the writing.
2
votes
1answer
49 views

Is the quotient $R/(a,b)$ equal to first quotienting $R$ with $(a)$ and then with $(b)$

Is the quotient $R/(a,b)$ equal to first quotienting with $(a)$ and then with $(b)$? I've been thinking about this for some time. And I think the following is true: ...
2
votes
1answer
259 views

Describe the rings: a) $\mathbb{Z}[x] / (x^2 - 3, 2x + 4)$, b) $\mathbb{Z}[i]/ (2 + i)$ [duplicate]

Describe each of following the rings: a) $\mathbb{Z}[x] / (x^2 - 3, 2x + 4)$ b) $\mathbb{Z}[i]/ (2 + i)$ a) Well, $\mathbb{Z}[x]$ is the set of all polynomials with integer coefficients and ...
0
votes
1answer
121 views

Let $I, J$ be ideals in a ring $R$. Prove that the residue class of any element of $I \cap J$ in $R/IJ$ is nilpotent.

Let $I, J$ be ideals in a ring $R$. Prove that the residue class of any element of $I \cap J$ in $R/IJ$ is nilpotent. So I know that an element $x$ of a ring is nilpotent if some power of $x$ is ...
1
vote
1answer
113 views

a question about a ring (unit, multiplicative inverse)

S is the power set of integers Z. Define two binary operations: $\bigoplus $: $A$\bigoplus $B=(A\bigcup B)-(A\bigcap B)$, the symmetric difference set $\ast : A \ast B = A \bigcap B$, which forms a ...
2
votes
1answer
83 views

Let R = Z[X]. Show that: I = {n + XP : $n\in2Z$, $P\in R$} is an ideal of R and that it is not a principal ideal

Let R = Z[X]. Show that: I = {n + XP : $n\in2Z$, $P\in R$} is an ideal of R and that it is not a principal ideal. i know what the ideal and principal ideal means but get stuck when proving it ...
1
vote
1answer
132 views

Ring of polynomials

Let F be a field and R = F [X], the ring of polynomials over F . Show that $$R^X=F^X$$,the set of non-zero constant polynomials. I am having a little trouble first understanding the question hence ...
2
votes
1answer
34 views

Show that $R[X,Y]/(X^2,Y) = R[X]/(X^2)$

I'm trying to show that $R[X,Y]/(X^2,Y) = R[X]/(X^2)$. I tried this: $$R[X,Y]/(X^2,Y)=(R[X])[Y]/(X^2,Y)=\frac{(R[X])[Y]/(Y)}{(X^2,Y)/(Y)}=\frac{R[X]}{(X^2,Y)/(Y)}\overset{?}{=}R[X]/(X^2)$$ I know ...
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
168 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
86 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
159 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
65 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
603 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
79 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
95 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, ...
1
vote
1answer
52 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
334 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 $R$ ...
-4
votes
2answers
96 views

Is $\mathbb{Z}$ a commutative ring?

Is $\mathbb{Z}$ a commutative ring? If so, would this imply that $\phi:\mathbb{Z}\to\mathbb{Z}$ is a commutative ring isomorphism? I know that $\phi$ is an isomorphism. I just don't know if it is ...
1
vote
2answers
167 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
402 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
86 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
221 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
105 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
243 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
61 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
443 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
73 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
61 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
50 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
156 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
146 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
93 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
63 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
35 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
497 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 ...