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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1answer
85 views

$p-1$ divides ord($x$)

Let $x$ be a primitive root modulo $p$. I have to prove that $p-1$ divides ord($x$) where $x\in\mathbb{Z}/p^n\mathbb{Z}$. I had to prove the following results before this (in which I succeeded): ...
2
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1answer
69 views

Is $(ad-bc,a+d)$ a prime ideal in $k[a,b,c,d]$ where $k$ is a field?

I need to show whether this is true: Let $k$ be a field and $k[a,b,c,d]$ be a ring of polynomials. Then the ideal $(ad-bc,a+d)$ is prime. If it is true, how should I proceed? Thanks!
5
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1answer
183 views

Rings in which every irreducible ideal is primary

Suppose $R$ is a commutative ring with $1$. It is well-known that if $R$ is Noetherian, then every irreducible ideal is primary (Lemma 7.12 in Atiyah & Macdonald). Is the converse true? That is: ...
2
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1answer
108 views

Why is $\mathbb{F}_p[X]/(X^2+1)$ not a field if $p \equiv 1 \bmod4$

here I am again with another question. Assume that p is prime, and $ \ p \equiv 1\bmod4$. Prove that $\mathbb{F}_p[X]/(X^2+1)$ is not a field. I don't know how to tackle this problem. First I'll ...
1
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1answer
94 views

$I=\{f\in \mathbb{C}[x,y,z,t] : f(-2,-1,1,2)=0\}$. Generators for $I$?

This problem is from a past qualifying exam I am trying to work on. I am stuck on trying to find generators for $I$. The question is as follows Let $\mathbb{C}[x,y,z,t]$ be the polynomial ring ...
6
votes
1answer
144 views

Are there Infinite Quotients of Algebraic Extensions of $\mathbb{Z}$?

It is well known that $\mathbb{Z}[a_1, \dots, a_n]/(a)$ is a finite ring if each $a_i$ is an algebraic integer and $a \neq 0.$ I suppose this statement becomes wrong if we just require those ...
1
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1answer
194 views

Example of semiprime ring

The ring is semiprime if $x\in R, xyx=0$ for all $y\in R$ implies $x=0$ or equivalently for $x\neq0$ exists $y_{0}\in R$ such that $xy_{0}x\neq0$. I found an example of semiprime ring. However, I am ...
8
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1answer
671 views

Irreducibility check for polynomials not satisfying Eisenstein Criterion.

My Question is to check Irreducibility for polynomials not satisfying Eisenstein Criterion. As an Illustration, to check whether $x^{p-1}+.....+x+1$ for p a prime is irreducible or not, we replaced ...
2
votes
1answer
513 views

What's are all the prime elements in Gaussian integers $\mathbb{Z}[i]$

So far, I know if $p$ is a rational prime, then $(1)$ if $p\equiv 3\mod4$, then $p$ is prime in $\mathbb{Z}[i]$. $(2)$ If $p\equiv1\mod4$ then $p=π_1 π_2$ where $π_1 $ and $π_2$ are conjugate, Then ...
1
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1answer
77 views

Maximal ideal $M$ such that $\Bbb Z[x] /M \simeq \Bbb Q$

Consider the ring $\Bbb Z[x]$. Does there exist a maximal ideal $M$ such that $\Bbb Z[x] /M \simeq \Bbb Q$ ? I don't think so, because $\Bbb Z[x]$ is not field.
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3answers
105 views

Direct Product, and Subdirect Product in Herstein text

I'm reading Noncommutative Rings by I. N. Herstein. And I find one lemma pretty strange. It's on page 52 of the book. I'm typing everything necessary all here, so everyone can have a look at it. ...
1
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1answer
178 views

In an UFD which of the following ideals is prime?

Let $R$ be a unique factorization domain and let $a,b\in R$ be distinct irreducible elements. Could anyone tell me which of the following is true? $\langle 1+a\rangle$ is a prime ideal. ...
8
votes
3answers
465 views

Error in proof: $\mathbb{C} \cong \mathbb{C} \times \mathbb{C}$??

I've unintentionally "proved" the following: $$\mathbb{C} \cong \mathbb{C} \times \mathbb{C}$$ Can you help me tracing the error I made resulting to this non-proof? Here it is. First of all, I recall ...
4
votes
1answer
78 views

If $A, B, C$ are finitely generated $\mathbb{Z}/{p^n}\mathbb{Z}$-modules such that $A \oplus B \simeq A \oplus C$, then $B \simeq C$

A friend that is preparing for an algebra qualifying exam asked me the following question yesterday, but I really have no idea of how to approach the problem. Let $A, B$ and $C$ be finitely ...
5
votes
1answer
175 views

A commutative ring in which every prime ideal is 2-generated

Suppose $R$ is a commutative ring with 1. There are some statements that tells us if prime ideals behave in certain way, then all the ideals will behave in that way. For example, If every prime ...
2
votes
1answer
62 views

If $R$ is a ring, $x^3\in Z(R)$ and $x^2\in Z(R)$ then $x\in Z(R)$ . [closed]

Is this true and why? If $R$ is a ring, $x^3\in Z(R)$ and $x^2\in Z(R)$ then $x\in Z(R)$ .
2
votes
1answer
178 views

Let $R$ be a ring satisfying $(xy)^3=xy$ for all $x,\, y\in R$. Then $R$ is commutative.

Let $R$ be a ring satisfying $(xy)^3=xy$ for all $x,\, y\in R$. Then $R$ is commutative. Any suggestion how to prove it?
0
votes
1answer
110 views

If [x,z] and [y,z] commute for all x,y,z in R, then $[x,y]^4=0$ for all x,y in R

If $\bigl[x,\, z\bigr]$ and $\bigl[y,\, z\bigr]$ commute for all $x,\, y,\, z\in R$, then $\bigl[x,\, y\bigr]^{4}=0$ for all $x,\, y\in R$. In such a ring $R$ a commutator $c$ satisfy the equation of ...
2
votes
1answer
122 views

If the group of units of a unital ring is cyclic, must it be finite?

Suppose you have a ring $R$ with $1$ and that the multiplicative subgroup $R^\times$ of $R$ is cyclic. Is then $R^\times$ a finite group? Are there conditions, such that $R^\times$ is cyclic, or is ...
2
votes
2answers
250 views

Elements which don't have a $\gcd$ in $\mathbb Z[\sqrt{-5}]$

Show that in the ring $R = \{a + b\sqrt{-5} |\, a,b\,\, \in \,\mathbb{Z} \}$ the elements $\alpha = 3$ and $\beta = 1 + 2\sqrt{-5}$ are relatively prime but $\alpha\gamma$ and $\beta\gamma$ have ...
6
votes
2answers
116 views

If $\gcd(f(x), g(x))\ne1$, then $F[x]/(fg)$ is not isomorphic to $F[x]/(f)\times F[x]/(g)$

So I thought up this question as an extension to the corresponding one for $Z_m$, $Z_n$, $(m, n)\ne1$. The problems is I am unable to prove it, or disprove it. I try, but I keep getting tripped up ...
2
votes
1answer
43 views

how different is the notion of an “indeterminate” from that of “algebraically independent” in relation to dimension theory?

The following is a well-known theorem in commutative algebra, see e.g. Matsumura, Commutative Ring Theory, p. 117: Let $A$ be a Noetherian ring and $X_1,\cdots,X_n$ indeterminates over $A$. Then ...
3
votes
3answers
165 views

Are the unitary homomorphisms $\mathbb{R} \rightarrow \mathbb{R}$ and $\mathbb{C} \rightarrow \mathbb{C}$ the identity?

I have some trouble with an algebraic excercise about unitary mappings. First part If $f: \mathbb{R} \rightarrow \mathbb{R} $ is unitary, how to proof that $f$ can only be the identity? My ...
5
votes
4answers
1k views

Characteristic of an integral domain must be either $0$ or a prime number.

Proposition: Characteristic of an integral domain must be either $0$ or prime number. I'm confused by this proposition. I think the characteristic of an integral domain should be always $0$. ...
3
votes
1answer
298 views

Laurent Power Series - Polynomial Long Division

I am curious about making a laurent series for a rational function, if possible by long division of polynomials. For example, $$\frac{3}{x+2}.$$ If I do long division here, I'd first multiply the ...
11
votes
2answers
495 views

Number of elements in the quotient ring $\mathbb{Z}[X]/(X^2-3, 2X+4)$

I had to calculate the number of elements of this quotient ring: $$R = \mathbb{Z}[X]/(X^2-3, 2X+4).$$ This is what I've got by myself and by using an internet source: Writing the ring $R = ...
1
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2answers
119 views

Sum of ideals in the polynomial ring

Could someone explain to me how to find a sum of ideals where $I=(x+y)$ and $J=(x)$? The answer to this is $I+J=(x,y)$ and we work in the polynomial ring $k[x,y]/(xy)$. I know that the definition ...
1
vote
1answer
46 views

Ideals of nested PID's

Let $R\subset K$ be principal ideal domains. If $a,b$ are nonzero elements of $R$, prove that $I=J\cap R$, where $I$ and $J$ denote the ideals generated by $a,b$ in $R$ and $K$, respectively. Showing ...
2
votes
2answers
91 views

Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
9
votes
2answers
257 views

Cosets modulo $(2+i)$ in $\mathbb{Z}[i]$

I've been trying to solve this problem but I got stuck on it: Given is the ideal $I=(2+i)$ in the ring of Gaussian integers $\mathbb{Z}[i]$. How many elements does the quotient ring ...
3
votes
2answers
185 views

Example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable ...
3
votes
1answer
64 views

proving isomorphism of two $k$-algebras

Let $k$ be a field. I would like to prove that $k[x,y]/(x^3-y^2) \cong k[t^2,t^3]$. Of course, intuitively, i can readily see that this must be the case. More formally, i define a homomorphism ...
3
votes
4answers
190 views

Let $R$ be a ring that has no nonzero nilpotent commutators. If $e\in R$ is an idempotent, then $e\in Z(R)$.

Let $R$ be a ring that has no nonzero nilpotent commutators. If $e\in R$ is an idempotent, then $e\in Z(R)$. I have a problem with proving this theorem. I don't know how to understand nonzero ...
3
votes
2answers
190 views

If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.

If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative. I need to proof this theorem and I have something like this below. However, I do not know how to continue this proof. ...
3
votes
3answers
233 views

Order of $\mathbb{Z}[i]/(1+i)$ [duplicate]

I have to calculate the order of the ring $\mathbb{Z}[i]/(1+i)$. This is how far I am: If $a+bi\in \mathbb{Z}[i]/(1+i)$ then there are $n,m\in \mathbb{Z}$ such that $a+bi\equiv 0+ni$ or $a+bi\equiv ...
9
votes
4answers
541 views

$\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$

I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$. I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be ...
0
votes
3answers
95 views

How to prove that a particular ideal is the kernel of a ring homomorphism?

I have to prove that $(1+3i)$ is the kernel of the homomorphism $f:\Bbb{Z}[i]\to \Bbb{Z}/10\Bbb{Z}$ defined by $f(a)\to a \mod 10, a\in \Bbb{Z}$, and $f(i)\to 3 \mod 10$. I know that $(1+3i)$ is ...
17
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1answer
444 views

When is a group ring an integral domain

If $R$ is an integral domain (I am having $\mathbb{Z}$ or a field in mind) and $G$ a (not necessarily finite) group then we can form the group ring $R(G)$. Note that if $g^{n+1} = e$ then ...
1
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2answers
115 views

Presentation of finite rings (fields)

One knows that every finite group is isomorphic to a subgroup of $\operatorname{GL}(n)$ for some $n$ large enough. Can every finite ring be represented by a ring of matrices, i.e., is every ring ...
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votes
3answers
177 views

$(-a)^2=a^2$ in commutative ring?

Maybe this is a silly question, but how can I show that $(-a)^2=a^2$ in a commutative ring with $1$ for all $a$ in the ring? I know that $(-a)^2=(-a)\cdot(-a) =(-1)\cdot(-1)\cdot a^2$. So I ...
2
votes
2answers
310 views

What does Herstein mean by 'centroid of a ring'?

I'm currently reading Herstein's Noncommutative Rings, and the definition of the centroid of a ring is on page 46 of the book. Let $\text{End}(R)$ be the ring of endomorphisms of the additive group ...
2
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1answer
57 views

An advanced algebra question

How can I show that $\Bbb{R}(x)$ (the quotient field of $\Bbb{R}[x]$) is not a real closed field ?
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0answers
83 views

Non Classical Examples of Indecomposable Ideals

A classical example of a ring $R$ with an indecomposable ideal is the ring $C(X)$ of real valued continuous functions on $X$, where the $(0)$ ideal is not decomposable. Does anyone know other examples ...
1
vote
1answer
101 views

Direct sum of commutative rings

Let $R$ be a direct sum of ideals $R=R_1\oplus R_2\oplus\dots\oplus R_k$. Each ideal $R_i$ is commutative of order $p_{i}^{n_{i}}$ ($p$ is prime), and has a unity. How to show that the direct sum of ...
5
votes
1answer
87 views

Simple + Artinian = Semiprimitive

By a noncommutative ring I mean that it has no unit. I know that if some ring (say, $R$) is simple, then: $R^2 \neq (0)$ It only possesses $2$ two-sided ideals, namely $(0)$, and itself. And ...
8
votes
6answers
516 views

Examples of a commutative ring without an identity in which a maximal ideal is not a prime ideal

In a commutative ring with an identity, every maximal ideal is a prime ideal. However, if a commutative ring does not have an identity, I'm not sure this is true. I would like to know the ...
2
votes
4answers
152 views

Yet another characterization of the field $\mathbb{Z}/2\mathbb{Z}$

Let $R \neq 0$ be a ring which may not be commutative and may not have an identity. Suppose $R$ satisfies the following conditions. 1) $a^2 = a$ for every element $a$ of $R$. 2) $R$ has no two-sided ...
0
votes
1answer
178 views

Find all homomorphisms

Find all ring homomorphisms $\Phi$: $\mathbb{Z}_2 \rightarrow \mathbb{Z}_6$ and $\Phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2$.
2
votes
1answer
54 views

Properties of quasiregular elements in a matrix ring

I've been puzzling over one of the properties of quasiregular elements listed in the wikipedia article on the topic. An element of a ring $x$ is quasiregular (left, right) when $1-x$ has a ...
-1
votes
2answers
115 views

Ring of order $p^2$ and its characteristic

I suppose that this question might be very easy for some people. However, I have got problem to get it. Could anyone explain to me why the characteristic of a finite ring of order $p^2$ is $p$. I know ...