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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Invertible element of $S$

Let $S=\mathbb{Z}[\sqrt{2}]$ = {$a+b\sqrt2|a,b\in \mathbb{Z}$} and $R = \mathbb{Q}[\sqrt2]$ = {$\alpha + \beta\sqrt2 | \alpha, \beta \in \mathbb{Q}$}. Consider $x=3+2\sqrt2$ and $y = 3+4\sqrt2$ ...
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Integral extension is a finitely generated $R$-module?

Let $R$ be a commutative ring. If $b_1,\ldots,b_n$ are elements of a ring $R'$ (commutative) which are integral over $R$ then $R[b_1,\ldots,b_n]$ is a f.g. $R$-module. My question is: If ...
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2answers
36 views

Ideal as kernel of a homomorphism in Gaußian integers

Consider the ring $\mathbb{Z}[i]$ of Gaußian integers. The principal ideal $(1+i)$ is maximal ideal in this ring. Since ideals are kernels of some homomorphisms, I would like to see a homomorphism ...
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Is simple module over commutative ring always a field?

M is a simple module if and only if $M\cong R/I$ for some I maximal ideal in R. If $R$ is commutative, can I say M is a field? I'm confused about this fact because when proving it I use the fact that ...
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0answers
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If α is a unit in $\mathbb{Z}[\sqrt2]$ then $a^2 - 2b^2 = \pm1$ [duplicate]

$$\mathbb{Z}[\sqrt2] = \{a + b√2: a,b \in \mathbb{Z}\}$$ The question is asking to prove that if $\alpha$ is a unit in this set, then: $$a^2 - 2b^2 = \pm1$$ I've hit a dead end already: I wanted ...
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1answer
21 views

If $R$ is a commutative simple ring with identity , then is any matrix ring $M_n(R)$ also simple? [duplicate]

If $R$ is a commutative simple ring with identity , then is any matrix ring $M_n(R)$ over $R$ of matrices of size $n$ also simple ?
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2answers
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Determine all units in $\mathbb{Z}[\omega] := \{a+b\omega\mid a,b\in\mathbf{Z}\}$ where $\omega = \frac{-1 + i \sqrt{3}}{2}$

My attempt: $N(a + b\omega) = (a + b \omega)(a - b \omega) = a^2 + \omega^2 b^2$ I'm stuck here. Is my approach correct?
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1answer
22 views

For a maximal ideal $M$ of $R$ of a commutative ring $R$ ( not necessarily with unity ) , then is $R/M$ a simple ring ?

Let $M$ be a maximal ideal of a commutative ring $R$ ( not necessarily with unity ) ; then is it true that the only ideals of $R/M$ are the trivial ones i.e. is it true that $R/M$ is a simple ring ? ...
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26 views

$Rx$ is a simple module

Let $M\neq 0$ be an $R-$module. If $M=Rx$ for each $0\neq x\in M$, why is M a simple module?
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26 views

Help in understading this assertion..

The assertion says: Let $\psi : R \to S$ be a ring homomorphism. Composing $\psi$ with the inclusion $S$ as a subring of $S[x]$ obtain a homomorphism $\phi : R \to S[x]$. Then by [insert previous ...
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1answer
27 views

Let R be a ring and S be a subring of R with unity.

Let $R$ be a ring and $S$ be a subring of $R$. Suppose that $R$ does not have unity, but $S$ does. Let $1_S$ be the unity of S. Show that $1_S$ is a zero divisor of $R$. I've been stuck on this for ...
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2answers
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Why Can I divide generating function by $x$

In many books on generating functions author performs following operation to shift coefficients of $F(x) = \sum_i f_ix^i$ to the left $${F(x) - f_0} \over x$$ which in can be written as $$(F(x) - ...
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2answers
36 views

$I$ is maximal ideal $\implies$ $R/I$ has no proper ideals

I'm reading through a proof in a book on commutative algebra and in the proof it uses the fact that $I$ is a maximal ideal $\implies$ $R/I$ has no proper ideals, by using the correspondence theorem. ...
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1answer
41 views

$g(x) | f(x)$ show that $(f(x)) \subset (g(x))$

I have been given a problem recently that has been puzzling me for some time. The problem states If $g(x), f(x)$ are elements of a polynomial ring $F[x]$ and $g(x) | f(x)$ show that $(f(x)) \subset ...
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0answers
13 views

Cyclic group generator and Multicative identity iof Correspondng Ring

Can cyclic groups made into ring with unity such that multiplicative identity is not any generator?(Or does there exist example of one such cyclic group) Can we make (Z, +) into ring with unity ...
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What does an ideal generated by a subset look like?

I am acquainted with principal ideals relatively well but I am struggling to understand the more general definition of ideals generated by an arbitrary subset of a ring. I suppose my question comes ...
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1answer
19 views

Show that $V(\bigcup_{i \in I} E_{i})=\bigcap_{i \in I} V(E_{i})$

This is a part of a problem in Atiyah's Introduction to Commutative Algebra introducing the Zariski Topology. Here we are given that $(E_{i})_{i \in I}$ is a family of subsets of a unital commutative ...
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1answer
48 views

Maximal ideal in local ring

The maximal ideal in $\mathbb{Z}_{(2)}$ should be $(2)$, but I don't understand this well. Suppose I take $\frac35\in \mathbb{Z}_{(2)}$. It is not in $(2)$ but in $(3).$ But what is the ideal between ...
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2answers
63 views

Infinite rings with lots of zero divisors

Today I was trying to find an infinite ring $R$ whose all nonzero and nonidentity elements were zero divisors and actually found one: $\mathcal R =\text{Fun}(\mathbb N, \mathbb Z/2\mathbb Z)$. Given a ...
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1answer
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Let $K$ be a field, $A \subset K$, and $p \subset A$. Then $\exists$ a valuation ring $R$ satistfying…

I was stuck when reading a proof of the following theorem (Matsumura p. 72-3, Theorem 10.2), Let $K$ be a field, $A \subset K$ a subring, and $p$ a prime ideal of $A$. Then there exists a ...
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1answer
22 views

Show the union of two subrings is generally not a subring

Show that the union of two subrings is a subring if and only if either of the subring is contained in the other. I have no trouble in going from right to left but cannot seem to be able to go from ...
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1answer
33 views

Simple and semi-simple over $\mathbb{Z}$

What is a necessary and sufficient condition on an integer $n$ for $\mathbb{Z}/n \mathbb{Z}$ to be simple as a module over $\mathbb{Z}$? Semisimple? In the case of simple I think that because it is a ...
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1answer
47 views

Positive Real Numbers forming a subring

I was wondering if the subset of positive real numbers forms a subring of the real numbers under the regular operations of addition and multiplication. My thought so far is that since 1 is clearly in ...
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47 views

A relation between the Jacobson radicals of a ring and those of a certain quotient ring

Let $R$ be a ring $J(R)$ the Jacobson radical of $R$ which we define for this problem to be all the maximal left ideals of $R.$ I'm trying to prove the following proposition with only the definition ...
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1answer
44 views

Ring with many one-sided zero-divisors

Does there exist a ring all of whose elements are left zero-divisors but only one element is a right zero-divisor? The motivation for asking this question is that if there exists atleast one left ...
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1answer
39 views

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R.

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R. My attempt: Let $a \in S$. Then $1_S*a = a$ and this a ...
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1answer
30 views

Vector spaces and multiplicative inverse?

Do vector spaces have multiplicative inverses? They seem to be monoids under $+,\times$, so monoids $(\Bbb F, +)$ and $(\Bbb F, \times)$ where $\Bbb F=\Bbb R \,or\, \Bbb C$ And it is even a group ...
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Example of a Non-Commutative Division Ring With Finite Characteristics

Wedderburn's Little Theorem says that every finite Division Ring is commutative. What is about an infinite Division Ring with prime characteristics? Is this also a Field?
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Ideals of formal power series ring

I need help understanding the following solution for the given problem. The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i ...
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0answers
23 views

Prove: given a ring R with left identity $e_l$ and the right identity $e_r$, then $e_l = e_r$. Another way to prove?

Suppose a ring R has the left identity ($e_l$) and the right identity ($e_r$). Then $e_l = e_l*e_r = e_r$. I was wondering if there's another way to do it. Thank you.
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1answer
31 views

Why is it not a sufficient condition to conclude that a is a unity based only on the information that $xa = x$ for all $x$ in $R$?

We have a ring $R$ as follows: Why is it not enough to conclude that $a$ is a unity if $xa = x$ for all $x$ in $R$? Is it because it is by definition that the unity satisfies $ax = xa = x$ for all ...
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1answer
48 views

Does Euclidean division not work for general polynomials?

If $K$ is a field. Then in $K[X]$ there is an Euclidean algorithm and if $K$ is replaced by any arbitrary commutative ring $R$, then almost we have an Euclidean algorithm, by the following result: ...
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1answer
22 views

Commutative matrix question

I was doing my HW, and I am confused with one thing. To show that a matrix is commutative, do we need to show both $x+y = y+x$ and $xy=yx$? Or just by showing $xy=yx$ would suffice?
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Why we throw away the units in the definition of irreducible elements?

In the book "Abstract Algebra" by Dummit, the definition of irreducible element in an integral domain $R$ goes like this. Suppose $r\in R$ is nonzero and is not a unit. Then $r$ is called irreducible ...
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1answer
49 views

Can every group be extended to ring with idenity [duplicate]

Can every abelian group converted into ring(by defining multiplication operation) with identity with same order. We can convert every group G into ring by defining a.b = 0 for all a and b in G. But ...
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0answers
22 views

An integer $m$ is a prime element in $\Bbb Z[i] $ if $m$ is a prime number of the form $4n+3$ [duplicate]

An integer $m$ is a prime element in $\Bbb Z[i] $ if $m$ is a prime number of the form $4n+3$. I am stuck with the proof....please help!
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1answer
28 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
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2answers
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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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0answers
35 views

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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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
28 views

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
21 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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23 views

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

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
36 views

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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43 views

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
31 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
54 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
36 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
29 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 ...
2
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1answer
38 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 ...