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
28 views

Polynomial ring equals sum of sets.

Let $\mathbb{K}[x]$ the ring of polynomials in x with coefficients in $\mathbb{K}$. Let $$V_n = \left [nx^n + (n-1)x^{n-1} + \ldots + 1 \right ] $$ Show that $$\mathbb{K}[x] = ...
3
votes
2answers
94 views

Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. ...
0
votes
1answer
60 views

Abstract Algebra: Ring Homomorphism injective

Reference: Ring Homomorphism $\phi:f\to S$ is injective Referring this I have a doubt which I needed to clear. Below is my answer and query. We know $\phi:f\to S$ be ring homomorphism, where $f$ is a ...
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2answers
43 views

Properties of $R/I$

Let $R$ be an integral domain, and let $a$ be an irreducible element of $R$. Let $I$ be the ideal of $R$ generated by $a$. 1.If $R$ is a principal ideal domain, $R/I$ is a field ? True. Since $a$ ...
0
votes
1answer
47 views

Is the property of Euclidean domain inherited via surjective ring homomorphism? [duplicate]

Let $f:R \to S$ be surjective ring homomorphism and $R,S$ be integral domains. Could anyone advise me on how to prove/disprove this statement: If $R$ is Euclidean domain, then $S$ is Euclidean domain. ...
2
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0answers
43 views

A questions about Group Rings

Let's say $R:=\mathbb{Z}_p[C_{p^\infty}]$ be the group ring of a Prufer group over the field of integer module a prime $p$. We have $C_{p^\infty}=\langle u_1, u_2, ..., u_n, ... |\,\,\,\, ...
1
vote
1answer
44 views

Ascending chain condition and ring homomorphism

Let $f : R \to S$ be a surjective ring homomorphism between two integral domains. Could anyone advise me on how to prove/disprove the following statements: If $R$ satisfies the ascending chain ...
2
votes
2answers
30 views

Let $R$ be integral domain and $r \not | a.$ If $r$ is prime and $r^k|ab,$ then $r^k|b ?$

Let $R$ be an integral domain and $a,b,r \in R.$ Let $r$ be prime. Suppose there exists positive integer $k$ such that $r^k$ divides $ab$ and $r$ does not divide $a.$ Could anyone advise me on ...
2
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0answers
39 views

Help needed in understanding a proof

Claim: Let $M$ be a $R$-module ($R$ is an integral domain) and $p \in R$ be a prime. Suppose there exists non-empty finite subsets $B$ and $C$ of $M \backslash\{0\}$ such that $M= \bigoplus_{m \in ...
0
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0answers
20 views

GCD and LCM Property

Let D = $\mathbb R + X\mathbb C[X]$ Show that $GCD(X, iX) = \mathbb R^\times$ and $LCM(X, iX) = \emptyset$ I have an outline of what to do but don't exactly know who to show all of it... First, ...
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0answers
10 views

Domain GCD Property

Let D be a domain and $\emptyset \subset A \subseteq D^*$ $d \in GCD(A)$ if and only if (d) is a minimum among the principal ideals containing (A) If $d \in GCD(A)$ then d|a for all $a \in A$ and ...
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2answers
47 views

When $R$ is a subring of $R[\alpha]$

I am reading Artin's Algebra, Section 5 of Chapter 11. The chapter begins by saying that we want to find a suitable ring extension $R[\alpha]$ that contains $R$ as a subring as well as $\alpha$. He ...
2
votes
3answers
37 views

An identity in Ring of characteristic $p$ prime

Is it true that in a ring of prime characteristic $p$ results that $(x-1)^{p-1}=1+x+x^2+...+x^{p-1}$ ? If this is not true in general, the assumption that $x$ is a nilpotent element (let's say ...
1
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0answers
27 views

How to show that the dimension of a quotient space in the field of polynomials is not finite?

I have to show that if I have a quotient of the form $\mathbb{K}[x_1,x_2,\dots,x_n]/\langle f_1,f_2,\dots,f_s\rangle$, $\operatorname{char}(\mathbb{K})\not=0$, and on which the class of $[x_i^l]$ is ...
2
votes
0answers
91 views

Associated primes and their heights

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal ...
1
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1answer
43 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
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3answers
100 views

Ring theory question: $I=\langle x,2 \rangle$ prime/maximal ideal in $\mathbb Z[x]$?

In $\mathbb{Z}[x]$ , let $I = \lbrace f(x) \in \mathbb{Z}[x] : f (0) \text{ is an even integer} \rbrace.$ Is $I=\langle x,2 \rangle$ a prime ideal of $\mathbb{Z}[x]$? Is $I=\langle x,2 \rangle$ a ...
0
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0answers
19 views

Rings in which left regular elements are right regular

Is there a characterization of those rings in which every left regular element is right regular?
7
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1answer
115 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
2
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1answer
86 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
0
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1answer
16 views

Property of GCD in ring

Let D be a domain and $\emptyset \subset A \subseteq D^*$ Show that CD(A)={$d\in D$ | $(A)\subseteq (d)$} I know that I'll need to show both containments to show that the two statements are ...
1
vote
1answer
22 views

Ring with an element that's a left zero divisor and has a right inverse?

I'm looking for a (necessarily noncommutative) ring with $1$ which contains elements $a$, $b$, and nonzero element $c$, such that $ab=1$ and $ac=0$. The only noncommutative rings I know of are matrix ...
-2
votes
1answer
67 views

Surjective Implies Injective for R-Homomorphism on Finitely Generated Module [duplicate]

Let $M$ be a finitely generated module over a ring $R$, and let $f$ be an $R$-homomorphism from $M$ to itself. Does $f$ injective imply $f$ surjective? Does $f$ surjective imply $f$ injective? I have ...
1
vote
1answer
46 views

When is $\mathbb{Z}[\sqrt{d}]$ not an UFD (for $d>1$)?

I was wondering if there is a classification for this: For which $d$ is $D=\mathbb{Z}[\sqrt{d}]$ are UFD, with $d > 1$? For $d \equiv 1 $ (mod $ 4$), $D$ is not an UFD (proof here).
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votes
1answer
10 views

GCD property of Domain

Let D be a domain and $\emptyset \subset A \subseteq D^*$ If $x \in D^*$ and $GCD(xA)\neq \emptyset$ then $GCD(A)\neq\emptyset$ and $GCD(xA) = xGCD(A)$. I've already figured out how to show that ...
0
votes
1answer
39 views

GCD Domain Proof

Let $D = \mathbb{R} + X \mathbb{C}[X]$ Show that $\gcd_D(X^2,iX^2)=\emptyset $ Here is my plan so far... (and my questions) Suppose $f \in \gcd_D(X^2,iX^2) $. How do I show that because X is ...
1
vote
1answer
44 views

Does existence of $\gcd$ implies PID?

We know that if $A$ is a PID, then we can guarantee the existence of a $\gcd$ between two elements. My question is, does the converse hold? If I know that every two elements have a $\gcd$, do I know ...
1
vote
1answer
41 views

Prime elements of $\mathbb{Q}$

I'm quite confused about the existence of prime elements in the ring $R=\mathbb{Q}.$ We know that $r \in R$ is a prime iff $r$ is a nonzero, nonunit of $R$ and $r|ab \implies r|a \ \text{or} \ \ r|b ...
2
votes
2answers
54 views

Irreducibililty of $X^6+X^3+1$ in $\mathbb{Q}[X] \ $

Could anyone advise me on how to prove $X^6+X^3+1$ is irreducible in $\mathbb{Q}[X] \ ?$ I'm thinking of substituting $X=Y+1$ into the equation, do some tedious computations to simplify and use ...
0
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0answers
20 views

Operations in a polynomial ring over $\mathbb{F}_5$

Let $f(x)=3x^2 + 4x + 2$ and let $g(x) = 2x + 3$. Perform the following operations in $\mathbb{Z}/5\mathbb{Z}[x]$. (a) $f(x) + g(x)$ (b) $f(x)g(x)$ (c) divide $f(x)$ by $g(x)$. What is the ...
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0answers
8 views

Question on a sentence of an answer on prime element and modulo $p$.

shreevatsar Posted this as part of an answer to another question and I did not want to take over that question with my questions about this one part of the answer. So I posted it another question. ...
0
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1answer
31 views

Finitely generated ideal in boolean ring [duplicate]

A boolean ring is a commutative ring where $x^{2} = x$ for every $x$. Why in such a ring a finitely generated ideal is principal ?
4
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3answers
200 views

Idempotent in a local ring

Is it true that a local ring, i.e. a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
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2answers
64 views

Zero dimensional local ring with maximal ideal not principal.

Probably it is well known. I am looking for a zero dimensional local ring with maximal ideal not principal.
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0answers
46 views

Are Ideals and Varieties Inclusion Reversing?

Let $S_1$, $S_2$ be sets or varieties (I don't think it matters, does it?). Then if $S_1 \subset S_2$, is it always the case that $I(S_2) \subset I(S_1)$ (where I is an ideal)? Also, is it always the ...
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2answers
44 views

Ring and Subring have different identity

$(R,+,\cdot)$ is a commutative ring with identity 1 and $(S,+,\cdot)$ is a subring with identity 1'. Prove that if $1\ne 1'$, then 1' is a zero divisor in $(R,+,\cdot)$.
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0answers
22 views

For which $d\in\mathbb{Z}$, $\mathbb{Q}(\sqrt{d})$ primitive root of unity of order $p>2$ prime

If $p>2$ is a prime number, then I have to find $d\in\mathbb{Z}$ such that we have a primitive root of unity of order $p$. I know that $d<0$ because otherwise, you can never have a root of ...
0
votes
1answer
18 views

If $rk J=0$ then $R$ is a field

If $R$ is a regular local ring with maximal ideal $J$ and $rk J=0$ why is $R$ then a field? I have this as a statement in my notes and I feel that it should be obvious but I have no idea why it is ...
4
votes
1answer
63 views

Localization of Coordinate Rings: $\mathbb C[V_f] = \mathbb C[V]_f$.

Let $V\subseteq\mathbb C^n$ be an irreducible affine variety, then the coordinate ring $$\mathbb C[V] = \mathbb C[x_1,\dots,x_n]\big/\mathbf I(V)$$ is an integral domain. Let $f\in\mathbb ...
6
votes
1answer
67 views

$S^{-1}(\mathbb{Z}[i])$, where $S=\{x\in \mathbb{Z}|5\nmid x\}$.

Let $S=\{x\in \mathbb{Z}|5\nmid x\}$. I would like to know all the prime ideals of $S^{-1}(\mathbb{Z}[i])$. My attempt: Since $S\subset \mathbb{Z}$, the given question can be rewritten as ...
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3answers
73 views

An example of a noncommutative PID

It's well known that when a ring $R$ is a PID, every submodule of a free $R$-module is free. I'm interested in cases when the converse holds -- that is, in rings $R$ which have the property that every ...
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0answers
27 views

Isomorphism of Rings

True or False Let $f:R\to S$ be a homomorphism of rings. Then $f(1_R) = 1_S$ Let $f:R\to S$ be a homomorphism of rings. Then $f(0_R) = 0_S$ Suppose $u$ is a unit in a ring $R.$ The $u$ cannot be ...
5
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1answer
141 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
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2answers
34 views

Irreducible elements and Associates

Show that, in a domain, every associate of an atom is an atom. An atom is the same thing as an irreducible element. I think these two facts will be important to prove this statement: A nonunit ...
3
votes
1answer
62 views

Structure Theorem For PIDs

So, I'm a biologist at KCL, but I quite like mathematics and so am going through a book of exercises in algebra. Unfortunately, I've run into a problem in trying to answer some of the questions. I've ...
1
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2answers
40 views

prove that $ Z \left(\frac{1+\sqrt{m}}{2}\right)$ is noetherian ring.

prove that if m $\in \Bbb {Z}$ and $m\equiv 1\pmod 4$ , then $Z \left(\frac{1+\sqrt{m}}{2}\right)$ is noetherian ring. I need a help to do it . Thank you for your helping.
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1answer
30 views

Which power series with integer coefficients have multiplicative inverses?

Note that by this I mean Z[[x]], a commutative ring with 1 and integer coefficients. I've been racking my mind over this question. Considering the neutral multiplicative power here is 1 and thus is ...
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0answers
11 views

Why can an ideal generated by a subset be written in this form?

I have a subset $F \subset R$ that generates an ideal $(F)$. Apparently this can be written in the form $$(F)=\{a_1f_1b_1+...+a_kf_kb_k|k \ge 0, f_i \in F, a_i,b_i\in R\}$$ or if $R$ is commutative ...
0
votes
1answer
42 views

$X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X]$

Could anyone advise me on how to efficiently prove $X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X] \ ?$ Hints will suffice. Thank you.
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0answers
17 views

Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...