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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Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$.

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$. My Try: We can easily show that $\Bbb Z[i]$ is a FD but how can we show that $\Bbb Z[i]$ is a UFD. Because if we can show ...
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Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
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2answers
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Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$

I have to show that if $d > 2$ is a prime number, then $2$ is not prime in $\mathbb Z[\sqrt{-d}]$. The case when $d$ is of the form $4k+1$ for integer $k$ is quite easy: $2\mid ...
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1answer
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how does Macaulay2 computes analytic spread for non-local rings?

Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring. In the books like ...
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1answer
19 views

Is the (Krull) dimension of a semi-local Jacobson ring equal to zero? [duplicate]

Let $R$ be a commutative ring with identity element. If $R$ is semi-local (number of maximal ideals of $R$ is finite) and a Jacobson ring (this means that every prime ideal of $R$ is equal to the ...
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0answers
21 views

Units and ideals in the ring of integers modulo n [on hold]

a) How many units does the ring $Z_{60}$ have? Explain your answer. (b) How many ideals does the ring $Z_{60}$ have? Explain your answer.
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2answers
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Criteria for the existence of zero-divisors and idempotent elements in the integers modulo $n$

I need help in establishing or at least deciding the validity of the following two criteria: There are in the ring $Z_n$ non-trivial zero divisors if only if $n$ is divisible with some square. ...
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How to solve this algebra problem?

Let $e$ be the idempotent element of the ring R. If $\langle e\rangle$ is the principal ideal generated with $e$, show that $R\simeq\langle e\rangle\times A(\{e\})$. I think $A$ s ring which contains ...
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3answers
30 views

I need help to solve this problem

Let $R$ be a subring of a field $F$ such that for each $x \in F$ either $x\in R$ or $x^{-1} \in R$. Prove that if $I$ and $J$ are two ideals of $R$, then either $I \subseteq J$ or $J \subseteq I$.
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1answer
51 views

Automorphism group of the ring $\mathbb{F}_3\left[t,\frac{1}{t}\right]$

Let $R=\mathbb{F}_3\left[t,\frac{1}{t}\right]$ be a ring. What is the simplest form of $\mathrm{Aut}(R)$ ? Here $t$ is a variable and $R$ is the smallest ring contained in field ...
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1answer
21 views

Properties of Jacobson radical

I'm looking for an example of ring epimorphism $\varphi:R\rightarrow S$ such that the natural homomorphism $\tilde\varphi:J(R)\rightarrow J(S)$ is not surjective, where $J(R)$ is a Jacobson radical.
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1answer
25 views

Generators over semiperfect rings

It is clear that if $R$ is a ring with identity and $e\in R$ is an idempotent then $Re$ is a direct summand of $R$ while $R$ is a generator in the category of left $R$-modules. I have my question when ...
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1answer
34 views

Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$

Let $w\in\mathbb C$ be such that $w^3=1$ and $w\neq1$. Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$, and describe $\mathbb Z[w]/(2)$. What I wanted to do is to show that $\mathbb Z[w]$ is a ...
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2answers
52 views

Verify that R is a ring

Let $\alpha = \frac{1}{2}(1+\sqrt{-19}) \in \mathbb{C}$ and $R = \{a+b\alpha\mid a,b \in \mathbb{Z}\} \subseteq \mathbb{C}$. Verify that $R$ is a ring under usual addition and multiplication. My ...
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3answers
59 views

Simple form of a ring

What is a simple form of this ring: $$\mathbb{Z}[\sqrt{2}][x]/(5,x^2+1),$$ I know that $\mathbb{Z}[\sqrt{2}][x]=\mathbb{Z}[x,y]/(y^2-2)$. Probably, I should use second theorem of isomorphism, but I ...
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Rings With Bounded Index of Nilpotency are Dedekind Finite

Recently in an article by A. A. Klein I have seen this result: A ring $R$ with Bounded Index of Nilpotence is Dedekind Finite. Can anyone help me proving this result?
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1answer
58 views

Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence?

I have just started reading about the concept of $M$-regular sequences on my own and to understand the definition I asked myself the following question: Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence? ...
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1answer
40 views

What happens if we change the definition of quotient ring to the one that does not have ideal restriction?

From Wikipedia: Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if a − b is in I. ...
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1answer
51 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
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0answers
22 views

what happens if we adjoin elements in a ring not by ideals and quotient ring? [on hold]

We often adjoin elements in a ring by using ideals which results in a quotient ring. What happens if we adjoin elements that cannot use ideals method? What is the general property of the resulting ...
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3answers
61 views

Show that $R/I$ is a field, where $R$ is a PID , where $I$ is a nonzero prime ideal.

Let $I \neq \{0\} $ be a proper ideal of a $PID$ $R$ such that the quotient ring $R/I$ has no zero divisors. I have a problem in showing that $R/I$ is a field. Help Needed!!
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2answers
31 views

Show that $q\equiv_8 1$ when $q$ is an odd square number [duplicate]

Problem: Given: q is an odd squared number - show that: $q\equiv_8 1$ My assumption: $\forall q\in N:\exists a \in Z: a =1\pmod{2}$ and $a^2=q$. Then I tried to show that it's only true satisfyingly ...
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2answers
46 views

Why is $I$ often an ideal in quotient ring $A/I$?

When talking about quotient ring $A/I$, where $A$ is a ring, $I$ is often assumed to be an ideal. Why is this so? What makes ideals very important when discussing quotient ring?
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36 views

For a given integer $n>1$ , for which type of rings $R$ is it true that $(xy-yx)^n=0 , \forall x,y \in R \implies R$ is commutative?

For a given integer $n>1$ , for which type of rings $R$ is it true that $(xy-yx)^n=0 , \forall x,y \in R \implies R$ is commutative ? (It is obvious indeed that if $R$ is an integral domain or a ...
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2answers
51 views

Suppose $M$ is a Noetherian $R$-module and $\phi: M \rightarrow M$ is an $R$-module endomorphism of $M$.

So I did an exercise in my algebra textbook which was to show that $\ker(\phi^n) \cap \operatorname{im}(\phi^n) = 0$ and show that if $\phi$ is surjective, then $\phi$ is an isomorphism. I thought to ...
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1answer
38 views

For a commutative ring $R$, why does $1-ab$ being a non-unit leads to $1-ab \in M$ for some maximal ideal $M$?

Suppose there is a commutative ring $R$, without any restriction. Now suppose $a,b \in R$. If $1-ab$ is a non-unit, why is there at least one maximal ideal $M$ that $1-ab \in M$?
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Prove that 2 is reducible over $\mathbb{Z}[11]$ [closed]

Prove that 2 is reducible over $\mathbb{Z}[11]$.
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1answer
47 views

Is $(X^3 - 18X + 12, 5) \in \mathbb{Z}[X]$ a prime ideal?

I'm trying to determine wheter $A = (X^3 - 18X + 12, 5)$ and $B = (X^3 - 18X + 12, X-1)$ is a prime ideal in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I know that $A = \mathbb{Q}[X]$ since I can make ...
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1answer
37 views

For a non-unit element $x$ in a unital ring, does non-zero $a$ or $b$ ALWAYS exist s.t. $ax=xb=0$? [closed]

The question is as given in the title: For a non-unit element $x$ in a unital ring, does non-zero $a$ or $b$ always exist s.t. $ax=xb=0$?
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1answer
33 views

Is it always possible to extend a ring to a unital ring?

Just started learning algebra. So it's defined that ring is the ring not requiring a multiple 1, while unital ring does. Given a ring, is it always possible to extend it to a unital ring?
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1answer
42 views

Question related to commutative ring being Noetherian

Let $A$ be a commutative ring with $1$, and $A = (f_1, \ldots, f_n)$. I want to prove the following: If $A$ is a Noetherian ring, then so is $A_{f_i}$ (which is the ring $A$ localized at $f_i \in ...
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2answers
38 views

Is integrally closed domain finitely generated? [on hold]

Does integrally closed domains have finite number of generators that generate the whole ring?
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1answer
31 views

Is ring R itself a finitely generated module over $R$?

It seems trivial that ring $R$ itself is a $R$-module. But then can we say R is finitely generated by multiplicative identity? That seems so trivial..
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1answer
41 views

the number of zero divisors in polynomial ring

I was looking for an answer on the question How much zero divisors are in the ring $\dfrac{\mathbb{Z}_3[x]}{(x^4 + 2)}$? when I came up with the brilliant/hack-isch idea that it might just be ...
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0answers
26 views

Must local ring have multiplicative identity? [closed]

As the title says, must a local ring have multiplicative identity? Also must a regular local ring have multiplicative identity?
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2answers
55 views

Total ring of fractions of a Noetherian reduced ring is artinian

I'm doing the preparation to an exam, and I'm stuck in the following: If $R$ is a Noetherian ring with zero nilradical ($N(R) = 0$), and $S$ is the set of regular elements of $R$ ($r \in S$ if $rs ...
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1answer
33 views

Integral closed domain and localization of $\mathbb{Z}$ respect to prime ideal

We know that $\mathbb{Z}$ is integrally closed domain. This means that with respect to its prime ideal $p$, localization $\mathbb{Z}_p$ is also integrally closed in its field of fractions. Suppose ...
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2answers
64 views

integral ring extension, maximal ideals

Let $\varphi:A\rightarrow A'$ be an integral ring extension. 1) Show that for every maixmal ideal $m'\subset A'$ the ideal $\varphi^{-1}(m')\subset A$ is maximal 2) and that for every maximal ideal ...
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0answers
39 views

Counting rings of order $p^3$

MathWorld states that there are exactly 52 rings of order 8 (multiplication in rings may be not commutative and perhaps there will be no neutral element) and 53 rings of order $p^3$ where $p$ is an ...
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1answer
38 views

Characteristic of a ring: intuitive explanation

I know the following definition of characteristic of a ring: it is the smallest positive $n$ such that $$\underbrace{a+\cdots+a}_{n \text{ summands}} = 0$$ for every element a of the ring, if $n$ ...
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1answer
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example of an integral domain that is not integrally closed and showing that some localization is also integrally not closed

Can anyone show an example of integral domain that is integrally not closed and also show that one of its localization respect to maximal ideal is not integrally closed?
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How cannot localization of any integral domain respect to maximal ideal not be integrally closed?

Suppose that there is integral domain $I$. Now we take localization $I_m$ of $I$ respect to its maximal ideal $m$. $I_m$'s elements will consist of $a/b$ where $a \in I$ and $b \in m$. But integral ...
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2answers
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Conditions on ideal b for fields or integral domains

Let $A$ be a ring and $b$ be an ideal of $A$. Prove that 1. $A/b$ is a field $\iff b$ is maximal 2. $A/b$ is an integral domain $\iff b$ is prime I figure that the first is derived from the fact ...
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0answers
60 views

Solve $x^2 + 2 = y^3$ for integer $x$ and $y$ [duplicate]

I am asked to find all integers $x$ and $y$ which satisfy $x^2 + 2 = y^3$. I am given the hint that I should work in the unique factorization ring $\mathbb{Z}[\sqrt{-2}]$. So I could write the ...
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1answer
41 views

Why is $\mathbb{Z}_2[X]$ a principal ideal domain?

I used that $\mathbb{Z}_2[X]$ is a principal ideal domain to understand something but than I realized that in the lecture, we only noted that if we have a field, its polynomial ring is a domain. How ...
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2answers
55 views

Bijection between sets of ideals

Let $A$ be a ring and $\mathfrak{b}$ be an ideal of $A$. Prove that the assignment $$\mathfrak{c} \mapsto \mathfrak{c}/\mathfrak{b}$$ induces a one-to-one correspondence between the ideals of ...
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3answers
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Paradox of Field & Integral Domain in Venn Diagram

Check out venn diagram from this link, which is the order of number systems that has been embedded in my mind since grade school. Notice here that the integer $\mathbb Z$ is "inside" the rational ...
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3answers
95 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
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1answer
28 views

Ring isomorphism and indempotent element

Let $R$ be a ring. How to show that $R\cong R_1\times R_2 $, where $R_1,R_2$ are nontrivial rings, if and only if there exist $e\in R,\ e\neq0,1$ such that $e^2=e$ ? I need only hints.
3
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1answer
37 views

Isomorphism of rings (with parameters)

Let $p$ be a prime number. For each choice of $a,b\in \mathbb{F}_p,$ let $F(a, b)$ be the ring $\mathbb{F}_p[X]/(X^2+aX+b).$ Find all possible choices of $(a, b), (a', b')\in \mathbb F_p \times ...