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
0answers
85 views

$\mathbb{Z}[ \sqrt{−n}]$ is not PID [duplicate]

a) I prove that for $n=5$,$n=3$ that $\mathbb{Z}[ \sqrt{−n}]$ we have that 2 is irreducible but not prime, but how can i prove that in general for $n \geq 3$ $\mathbb{Z}[ \sqrt{−n}]$ 2 is irreducible ...
0
votes
1answer
33 views

Equivalence of zero divisor in commutative ring

Let $x$ be a nonzero element in a commutative ring, then $\exists y\neq0(xy=0)$ ($x$ is a zero divisor) iff $\exists y\neq 0(x^2y=0)$. $(\rightarrow)$ part is pretty trivial. How to prove the other ...
2
votes
0answers
297 views

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them. [duplicate]

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them. The above is the question, this is my attempt at an ...
2
votes
1answer
77 views

Prove that the elements $1, t-a, (t-a)^2, (t-a)^3,\dots, (t-a)^{n-1}$ form a $\mathbb{C}$-basis for the quotient ring $\mathbb{C}[t]/((t-a)^n)$.

Prove that the elements $1, t-a, (t-a)^2, (t-a)^3,\dots, (t-a)^{n-1}$ form a $\mathbb{C}$-basis for the quotient ring $\mathbb{C}[t]/((t-a)^n)$. $((t-a)^n)$ is the ideal generated by $(t-a)^n$. ...
3
votes
2answers
86 views

A non-UFD where we have different lengths of irreducible factorizations?

A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then ...
2
votes
1answer
83 views

Structure of $(\mathbb{Z}/p^k\mathbb{Z})^\times$ other than via $p$-adics?

I think I heard somewhere long ago that $\mathbb{Z}/p^k\mathbb{Z}$'s unit group is cyclic if $p$ is odd and $C_2\times C_{2^{k-2}}$ if $p=2$. I remember trying to prove it and finding it surprisingly ...
1
vote
1answer
62 views

What does multiplying an inverse of a quotient ring by a ring mean?

I am attempting to prove that a ring $R$ is the intersection of all $R_p$, where $R_p=S^{-1}R$ for S=R\P and the intersection is over all prime ideals P of R. The trouble right now is that I don't ...
3
votes
2answers
54 views

Prove that $ℤ[i]^*= \{1,-1,i,-i\}$

Prove that $ℤ[i]^*= \{1,-1,i,-i\}$. $\{1,-1,i,-i\} ⊂ ℤ[i]^*$ is trivial. But I'm not sure about the other inclusion. Let $(a+bi) \in ℤ[i]^*$. Then there exist $c,d \in ℤ$ such that $(a+bi)(c+di)=1$. ...
1
vote
2answers
101 views

Let $R$ a ring prove that $x(y-z)=xy - xz$

Let $R$ a ring (not necessarily commutative) prove that $x(y-z)=xy - xz$. \begin{align*} x(y-z)&=x(y+(-z)) \\ &=xy +x(-z) \\ &=xy+-(xz) \\ &=xy-xz \end{align*} I think all my steps ...
6
votes
4answers
258 views

Prime ideals in $C[0,1]$

Are there any prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\rightarrow \mathbb{R}$, which are not maximal? Perhaps, I duplicate smb's question, but this is an interesting problem! ...
0
votes
1answer
86 views

Characteristic $n$ and local rings

Prove that: a) If A is a local ring then A has characteristic zero or a power prime. Proof. Suppose M is the unique maximal ideal of A then $A/M$ is Field in particular integral domain then $Char( ...
1
vote
3answers
279 views

Suppose $\epsilon = a + b \sqrt d$ a unit in $R=\Bbb Z [\sqrt d] = \{a+b\sqrt d : a,b \in \Bbb Z\}$ . Proof that $\pm a\pm b \sqrt {d}$ are units.

I'm trying to solve some problems for my ring theory exam. I have problems solving this one: Let $d\in \Bbb Z_{>0}$ such that $d$ is not a square. Suppose $\epsilon = a + b \sqrt d$ a unit ...
1
vote
1answer
41 views

Existence of Algebra of anticommuting idempotents

Background and motivation: I'm wondering about the existence of an algebra which is in some ways similar to the exterior algebra, but is generated by idempotents rather than nilpotents. Let $V$ be a ...
2
votes
2answers
329 views

Let R be a commutative ring and let $A$ and $B$ be ideals of $R$. Show that if $A + B = R$, then $AB = A\cap B$

Let $R$ be a commutative ring and let $A$ and $B$ be ideals of $R$. (i) Show that if $A + B = R$, then $AB = A\cap B$ (ii) Now let $R$ be a Euclidean domain. prove that if $AB = A\cap B$, then $A + ...
1
vote
0answers
34 views

Grading on a free commutative graded algebra

Let $V$ be a one dimensional graded $\mathbb Q$-vector space; $V=\bigoplus_{i\geq 0}V_i$ and all $V_i$ are zero except $V_{2n+1}$ for some given $n$. Let $v$ be a generator of $V_{2n+1}$. Now take the ...
4
votes
1answer
54 views

Computing a quotient of rings

Let $R=k[x,y]/(y^2-x^2-x^3)$ and $I=(x,y)\cdot R \subset R$. I would like to show that $$ \bigoplus_{i=0}^{\infty} I^i\,/\,I^{i+1} \cong \,k[x,y]\,/\,(x^2-y^2). $$ Could you please help me? Remark: ...
1
vote
0answers
59 views

global dimension of rings and projective (flat) dimension of modules

Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite? Similarly, Let $R$ be ring such that ...
1
vote
1answer
142 views

Defining an ideal in the tensor algebra

In the wikipedia article about exterior algebra: The exterior algebra $Λ(V)$ over a vector space $V$ over a field $K$ is defined as the Quotient algebra of the tensor algebra by the two-sided ...
3
votes
2answers
121 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
130 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
65 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
77 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
36 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
81 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
94 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
146 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
300 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
137 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
143 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
97 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
146 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
37 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
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
100 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
168 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
69 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
765 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
83 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
106 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
435 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
175 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
465 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
93 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 $\{f(XY) \in \mathbb{R}[XY] : f \textrm{ has constant term 0}\}/(X^nY^m)\}$. ...