This tag is for questions about rings, which are an algebraic structure studied in abstract algebra and algebraic number theory.

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1answer
27 views

Are the generators of the subgroup defining tensor products linearly independent over $\mathbb Z$?

Let $S$ be a (commutative) ring with identity, and let $M$, $N$ be $S$-modules. (I guess if $S$ isn't commutative, I want $M$ to be a right $S$-module an $N$ a left $S$-module.) In the definition of ...
1
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0answers
70 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
2
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1answer
34 views

What is the difference between a module of finite rank and finitely generated module.

R is an integral domain and every module we talk about is an R-module. If a module is finitely generated then obviously every element of the module can be written as finite R-linear combination of the ...
3
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2answers
84 views

Is quotient of a ring by a power of a maximal ideal local?

Say I have a commutative ring $R$ with a maximal ideal $m$. Then $m/m^k$ is a maximal ideal in $R/m^k$ for any $k$. Is it the only maximal ideal, i.e. is $R/m^k$ a local ring? This is a well ...
1
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1answer
32 views

Example of excision in Hochschild homology

The excision theorem for Hochschild homology introduced by Wodzicki seems like a very powerful tool (as scision was hyper-useful in topology). However, I cannot actually seem to think of a result ...
4
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0answers
86 views

Can a division algebra over $\mathbb{R}^3$ be used to construct a counterexample to the hairy ball theorem?

Suppose that there is a (if necessary associative and/or normed) division algebra over $\mathbb{R}^3$. Is there a simple way to use this to construct a nonvanishing continuous tangent vector field on ...
1
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1answer
19 views

Does $I(J\cap K)=IJ\cap IK$ hold in a Dedekind ring?

For ideals in any ring, we have the relation $I(J\cap K)\subseteq IJ\cap IK$. Do we actually have equality if we are in a Dedekind domain? I've been looking around for a reference, but haven't found ...
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0answers
26 views

Question of a proposition about direct product

I try to prove it's injective, surjective and homomorphism. define f(x)=(x+a1,x+a2,....,x+an),it's homomorphism. it's injective <=> the intersection of ai=0 I don't know how to prove the ...
1
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1answer
20 views

Show polynomials $I$ is not finitely generated as $R$-module

Let $R=\{a_0+a_1X+\cdots+a_nX^n\;|\;a_0\in\mathbb{Z},a_1,a_2,\cdots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}_{\geq 0}\}$ and $I=\{a_1X+\cdots+a_nX^n\;|\;a_1,a_2,\cdots ,a_n\in\mathbb{Q}, n\in\mathbb{Z}^+\}$. ...
2
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3answers
71 views

Show $\ker (\phi)$ is a principal ideal

Let $\phi : \mathbb{C}[x,y] \rightarrow \mathbb{C}[t]$ be the ring homomorphism which satisfies: $\phi(x)=t^2,\ \phi(y)=t^2-t$ and $\phi(c)=c$ Show that the kernel of $\phi$ is a principal ideal. ...
-1
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1answer
50 views

Is this automorphism the identity map

Let $A$ be a commutative ring and let $f: A \rightarrow A$ an surjective homomorphism, let $a$ be a ideal of $A$ then if $f(a)\subseteq a$ then it's $f$ is the identity map, or not necessary.
1
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1answer
31 views

Example of an ordered, noncommutative division ring

Does there exist a noncommutative division ring $D$ (i. e. a field except that commutativity of multiplication is violated, e. g. the quaternions) which is also an ordered ring? Since most examples ...
1
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1answer
56 views

Let $R$ be a ring with $10$ elements, show that $R$ is commutative.

Let $R$ be a ring with $10$ elements, show that $R$ is commutative. $R$ is a ring which contains $10$ elements and doesn't have to include $1$.
0
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1answer
13 views

Prime ideals remain distinct

Let $A$ be a domain (commutative ring with unity and no zero divisors) and let $S$ be a multiplicative subset of $A$. Denote by $S^{-1}A$ the ring of quotients $\frac{a}{s}$ (you define the ring ...
2
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0answers
50 views

Proof for Unique Factorization Domain

Prove that the quotient ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a unique factorization domain. I am trying to prove first it is a principal ideal domain. However I am really stuck on this problem
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2answers
76 views

Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
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0answers
32 views

Question about singular homology

in order to prove that $H_0(X)\simeq \mathbb{F}$, $\mathbb{F}$ is the unitary commutative ring we have to prove that $C_0(X)/B_0(X)\simeq \mathbb{F}$ since we have that $C_0(X)$ is generated by the ...
1
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2answers
23 views

A question on Noetherian $R$ -module. [duplicate]

Let $M$ be Noetherian $R$-module(where $R$ contains $1$) and $\phi:M \to M$ be $R$ -module homomorphism . Suppose $\phi$ is surjective, how do I show that $\phi$ is injective ? Hints will suffice, ...
2
votes
1answer
46 views

$A_{p}$ is a field when $p$ is a minimal prime and $A$ reduced

$A$ is a reduced commutative ring with unit; $p$ is a minimal prime ideal. If $S = A \setminus{p}$ , I have to show that the ring $A_{p} = S^{-1}A$ is a field. My thoughts: Since $p$ is a minimal ...
3
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1answer
49 views

proposition 1.10 ii) A&M Introduction of commutative algebra

I am working through Introduction of commutative algebra and am having trouble with the following question: (I'll use f instead of the map,since I don't know how to input it.) Q1: Why there exist ...
1
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1answer
29 views

is there a counterexample of this map isn't surjective?

The ring A is a commutative ring with identity. I think ii) is true if they are not coprime. because for every (x+a1,....,x+an) we can find a x such that f(x)= (x+a1,....,x+an). Could you please ...
0
votes
1answer
13 views

showing $d(a)<d(ab)$ for $b\neq 0$ and $b$ not unit in Euclidean ring R.

Let $R$ be an Euclidean ring and $a, b \in R$. If $b\neq 0$ and $b$ is not unit in R, show that $d(a)<d(ab)$. Here is outline of the proof on my book. Let $A = (a) = \{ x a|x \in R\}$. Then ...
0
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0answers
24 views

Proof for uniqueness for ideal multiplication

I am across the following question here: The uniqueness of a special maximal ideal factorization Let R be a domain, and let I be an ideal that is a product of distinct maximal ideals in two ways, ...
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0answers
30 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where p is prime integer and f is primitive integer polynomial that is irreducible ...
1
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0answers
75 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
1
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1answer
31 views

Simple integral extension question

If $R$ is a commutative ring, why is every $x$ in $R$ integral over $R$? I can't see what monic polynomial will have $x$ as a root.
0
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1answer
16 views

ideal generated by irreducible polynomial

If $p(x)\in F[x]$ be a irreducible polynomial over $F$, what are the elements of $(p(x))$? From my book $(a) = \{ ra|r \in R\}$ but I don't know what $r$ is going to be in this case. I see ...
0
votes
1answer
59 views

Invertible elements and maximal ideals of a localization

Let $n\in\mathbb Z$ and let $A$ be the set of integers co-prime to $n$. Denote $A^{-1}\mathbb Z$ by $\mathbb Z_{(n)}$. 1) Find the invertible elements of $\mathbb Z_{(6)}$ My attempt: let $m$ be ...
0
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1answer
49 views

What does $R^{\times}$ mean for a ring?

What does it mean for $a \in R^{\times}$, where $R$ is a ring? I cannot seem to find the definition of $R^{\times}$ anywhere (partially because I do not know what it is called).
0
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1answer
38 views

Question about comaximal ideal proof

Let $A$ be a ring and $M\subseteq A$ a maximal ideal. Show that if $I\subseteq A$ such that $I\not\subseteq M$, then $M$ and $I$ are comaximal($M+I=A$). I cannot find the proof for this statement.
1
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1answer
44 views

Proof on unital ring isomorphism

I thought I proved something until I looked at the hints, which leads me to believe my proof attempt is too simple. $\phi:\mathbb{R}\rightarrow \mathbb{R}$ is a unital ring isomorphism (i.e. ...
0
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3answers
35 views

Show that $M_2(\mathbb{R})$ has no non-trivial two-sided ideals

In addition to the title question, I also want to find a non-trivial right ideal and a non-trivial left ideal of $M_2(\mathbb{R})$ . Attempt of title question: Suppose $\exists I\subset ...
3
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3answers
66 views

Ideal of a polynomial ring, and an isomorphism between $R[x]/I$ and $R$

Let $R$ be a ring. $I\subset R[x]$ is the ideal of all elements with a zero constant term. Show that $I$ is an ideal, and show that $R[x]/I\cong R$. Attempt: $R[x]=\{a_0+a_1x+...+a_nx^n:a_i\in R\}$. ...
2
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1answer
35 views

Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
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0answers
39 views

Ring $R$ as $R[x]$-module

My professor mentioned some interesting examples of modules, giving as an example the following two: $R$ as an $R[x]$-module, in which multiplication by $x$ was taken to be evaluation under a fixed ...
0
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1answer
49 views

Quotient field of a localization

I have a basic question about rings of fractions. Let $R$ be a commutative integral domain with quotient field $K$, $\mathfrak p$ a non-zero prime ideal of $R$ and $R_{\mathfrak p}$ the localization ...
0
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1answer
18 views

Definition of h.c.f./g.c.d. not fitting with $\mathbb{Z}$

In my lecture notes, and also on many websites, the definition of the highest common factor of two elements in an integral domain $R$, say $a$ and $b$, is an element $c$ such that: $c|a$ and $c|b$ ...
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1answer
47 views

How to find a chain of prime ideals in $\mathbb{Z}[x]$ [duplicate]

How can I build three prime ideals of $\mathbb{Z}[x]$, $P_1, P_2, P_3$ with $P_1 \subsetneq P_2 \subsetneq P_3$ and justify this?
0
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0answers
25 views

Linearly Independent and Span Proof

Let R be a field, M be an R-module, $X \subseteq M$. Show that $X$ is linearly independent if and only if $x$ not $\in$ span (X\ {x}) for each $x \in X$ (I'm not sure how to write the symbol for ...
0
votes
1answer
29 views

Span and Smallest Submodule Proof

Let R be a ring, M a R-module, and $X \subseteq M$ Show that span$(X)$ is the smallest submodule of R containing X. My ideas: Every submodule is contained in its span so $X \subseteq$ span$(X)$ and ...
3
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3answers
78 views

Examples of Commutative Rings with $1$ that are not integral domains besides $\mathbb Z/n\mathbb Z$?

I'm doing a problem where I'm trying to find a counterexample to some statement about commutative rings with $1$ when the ring is not a domain. I've tried looking at $\mathbb Z/n\mathbb Z$, for ...
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2answers
76 views

Showing that $R / I$ is a ring

Let $R$ be a ring and let $I$ be an ideal (that is, $I \le R$ and $R \cdot I \subset I)$. Define $$R / I = \{r + I : r \in R \}$$ Show that $R/I$ is a ring under the operations $(r+I) + (s+I) = ...
2
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2answers
54 views

“Primeness” of C[x] in B[x], where A is a subring of B and C is the integral closure of A in B.

Let A be a subring of B, and C the integral closure of A in B. If f, g are monic polynomials in B[x] such that fg is in C[x], then f, g are in C[x]. The first part of the problem allowed the ...
2
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1answer
54 views

Maximal ideals in $\mathbb{Z}[x]$

I am trying to solve the following problem from Artin: Every maximal ideal $\mathbb{Z}[x]$is of the form $(p,f)$ where p is a prime integer and $f$ is a primitive polynomial that is irreducible modulo ...
0
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2answers
35 views

Cardinalities of bases of a free $R$ module are same? [duplicate]

Let $R$ be a ring with no zero divisiors such that for all $r,s\in R$ there exist $a,b\in R$ not both zero with $ar+bs=0$. If $R=K\oplus L$ then $K=0$ or $L=0$. if $R$ has an idendity then any two ...
2
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1answer
30 views

Units in a polynomial ring

I'm trying to determine $U(\mathbb{R}[x])$, where $U(R)$ denotes the units group of a ring R. I think the answer is all non-zero constant polynomials, but I'm having trouble showing that these are ...
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0answers
33 views

Class Group of Ring of Integers of $\mathbb{Q}[\sqrt{-57}]$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need ...
1
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1answer
23 views

Homework: If a is a unit, prove that b divides c if and only if ab divides c.

As it says in the title, this is a homework question, so try not to give everything away. I'm just looking for a starting point. The question is stated as follows: Let $R$ be a commutative, unital ...
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2answers
33 views

Ring with special rules for add and mult

$R$ is a ring of integers with special rules for multiplication and addition. Suppose that $f: \mathbb Z \to R$ is an isomorphism that is defined by $f(x) = x-2$. What integer in the ring $R$ is ...
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0answers
34 views

Ring A is integral over the subring of invariants under a finite group action

I need to prove that if G is a finite group that acts on ring A, and $A^G$ is the subring consisting of elements of A which are invariant under all g in G, then A is integral over $A^G$. The hint ...