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
23 views

Ring with One sided Zero divisor [on hold]

Does there exist ring whose all elements are left zero divisor but only one element is right zero divisor
2
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1answer
32 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 ...
0
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1answer
26 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 ...
4
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2answers
46 views

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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2answers
29 views

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
20 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
27 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
45 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
20 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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2answers
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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 ...
3
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1answer
46 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!
1
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1answer
23 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 ...
2
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2answers
67 views

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 ...
1
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0answers
28 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 ...
0
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2answers
42 views

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 ...
1
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1answer
24 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 ...
1
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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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0answers
22 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
35 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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0answers
42 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
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
53 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 ...
1
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1answer
24 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
27 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
36 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 ...
2
votes
2answers
70 views

Verify that R is a ring

Let $\alpha = \frac{1}{2}(1+\sqrt{-19}) \in \mathbb{C}$ and $R = \{a+b\alpha\mid a,b \in \mathbb{Z}\} \subseteq \mathbb{C}$. Is R an integral domain with unity? My attempt: (Please correct me if I ...
3
votes
3answers
60 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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0answers
25 views

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? ...
3
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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. ...
1
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1answer
53 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 ...
2
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3answers
63 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!!
2
votes
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 ...
4
votes
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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0answers
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 ...
3
votes
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
39 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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0answers
26 views

Prove that 2 is reducible over $\mathbb{Z}[11]$ [closed]

Prove that 2 is reducible over $\mathbb{Z}[11]$.
2
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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 ...
1
vote
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. $ax=xb=0$?
2
votes
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
32 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..
2
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1answer
42 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
57 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 ...
3
votes
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 ...