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
54 views

Are there solutions to FLT which are linearly independent over $\mathbb{Z}$

Specifically, I would like to know if there is some $R$, where $R$ is a ring with unity $\mathbb{Z} \subseteq R$ there are $x,y,z \in R$ and a prime $p \in \mathbb{Z}$ such that $x^p + y^p + z^p = ...
1
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1answer
46 views

Definition of Ideal in a Ring

Let $R$ be a ring without unity. Let $I$ be a non-empty subset of $R$ satisfying: (1) $a,b\in I$ $\Longrightarrow$ $a+b\in I$. (2) $a\in I$ and $r\in R$ $\Longrightarrow$ $r.a\in I$. Is $I$ a ...
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0answers
27 views

betti-numbers of Gin(I), generic initial ideal of $I$

here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b): can you help please?
3
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2answers
60 views

Subring which is not an ideal?

Exercise: Give an example of a subring of a finite commutative ring $R$, that is not an ideal of $R$. I recently learned the following: Let $2^\Omega$ be the power set of an arbitrary set ...
2
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0answers
28 views

Quotients of successive powers of the augmentation ideal

For $H$ be any group, let $\mathbb{Z} H$ denote the integral group ring. Define $J(H)$ to be the augmentation ideal i.e $J$ is the kernel of the ring homomorphism $\mathbb{Z}H \to \mathbb{Z}$ sending ...
0
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1answer
58 views

On a complex group rings of finite groups

Let $G$, $K$ and $H$ be three finite groups where $H$ acts on both of $G$ and $K$ as sets nontrivially. Moreover assume that $\mathbb{C}K$ is (isomorphic to) a $\mathbb{C}H$-submodule of $\mathbb{C}G$ ...
0
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1answer
27 views

Examples of of non-commutative rings with no multiplicative identity ( finite and infinite ) other than matrix rings

Can anyone please give some examples or give a reference where I can find examples of non-commutative rings with no multiplicative identity other than matrix rings ? Also examples of finite ...
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1answer
22 views

For a (left) ideal $I$, how can $R \cdot I$ be a proper subset?

I have the following definition for a left ideal: An additive subgroup $I$ of a ring $R$ is called left ideal, if $R \cdot I \subseteq I$ where $R \cdot I = \{ r \cdot i \,\,\,|\,\,\, r \in R, i \in ...
2
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0answers
38 views

Generalisation of chinese remainder theorem on ideals of ring without 1

Let $I_1,\dots,I_n$ be (two-sided) ideals of a ring $R$ (not necessarily with 1), which are pairwise co-maximal, i.e. $\forall i\ne j\in \mathbb{Z}_{[1,n]}$, $I_i+I_j=R$. Let $f:R\to R/I_1\times ...
6
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2answers
56 views

Prove $\mathbb{Z}_3[x]/(x^2+1)$ and $\mathbb{Z}_3[x]/(x^2+x-1)$ are isomorphic.

Prove $\mathbb{Z}_3[x]/(x^2+1)$ and $\mathbb{Z}_3[x]/(x^2+x-1)$ are isomorphic by finding an explicit isomorphism. My question is how I can define the map. Here are what I tried: ...
4
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1answer
64 views

Ring of formal power series over a principal ideal domain is a unique factorisation domain

An exercise in my algebra course book asks to prove that if $R$ is a PID, then $R[[x]]$ is a UFD, where $R[[x]]$ is the ring of formal power series over $R$. After some failed attempts at proving the ...
0
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1answer
37 views

Let $p$ be a prime , then $\{0\} \cup\bigl\{\frac ab \in \mathbb Q : a \ne 0 \space ; (a,b)=1=(p,b) \bigr\}$ is a subring of $(\mathbb Q,+,.)$?

Let $p$ be a prime , then is the set $\{0\} \cup\Big\{\dfrac ab \in \mathbb Q : a \ne 0 \space ; (a,b)=1=(p,b) \Big\}$ a subring of $(\mathbb Q,+,.)$? The problem is, how can one show that this is ...
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0answers
36 views

Ring of integer valued polynomials is not Noetherian [duplicate]

Let $A := \text{Int}(\mathbb{Z}) :=\{ f \in \mathbb{Q}[x]: f(\mathbb {Z}) \subset \mathbb{Z} \}$. Why $A$ is a non-Noetherian ring ?
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1answer
30 views

A simple criterion for semi-simplicity

I made a simple observation about semi-simplicity of rings: A ring $R$ is semi-simple iff for every $x\in R$ there is an element $y\in R$ s.t. $xyx=x$. It seems very interesting to have such a ...
3
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1answer
40 views

Krull dimension of the quotient by a single element

Let $(R,m)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module of dimension $d$. The Krull dimension of $M$ is defined to be the Krull dimension of $R/\operatorname{ann}(M)$. ...
2
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1answer
40 views

Matsumura Exercise 6.3

The questions states: Let $A$ be a Noetherian ring and $x\in A$ be an element which is neither a unit nor a zero-divisor; prove Ass$_A(A/xA)=$Ass$_A(A/x^nA)$ for each $n=1,2,\ldots.$ My question ...
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0answers
59 views

What is the set of homomorphisms $\hom_\text{Ring} (\mathbb{Z}_{n},\mathbb{Z}_{m})$?

(For $A,B$ rings and $R$-modules, denote $\newcommand{\Hom}{\operatorname{Hom}}$ $\Hom_\text{Ring}(A,B)$ the ring of ring homomorphisms $A \to B$, contrasted with $\Hom_R(A,B)$ which is the R-module ...
1
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1answer
51 views

Does $a\mid b$ in a Euclidean domain imply $\operatorname{Norm}(b) \ge \operatorname{Norm}(a)$?

Is the following proposition true? Let $R$ be a Euclidean Domain, $a,b\in R$, $b\ne 0$, $a|b$. Then $N(b)\ge N(a)$. I cannot (in my very limited knowledge) think of any counterexamples, but I ...
3
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1answer
88 views

If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents

Let $R$ be a commutative ring with identity, $P$ be a prime ideal in $R$ and define $$X := \lbrace t \in P \mid t^2=t \rbrace. $$ Also let $J$ denote the smallest ideal of $R$ that contains $X$. ...
0
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2answers
36 views

How can the only maximal ideal of $C[x] / X^2$ be $(X)$?

In my notes I have the following example which I don't understand. Let $f$ be the canonical injection from $C$ to $C[X]/X^2$.The only maximal ideal of $C[X]/X^2$ is $(X)$ and $f^{-1}((X))$=$(0)$. ...
0
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1answer
47 views

Are Boo and BooRng really isomorphic?

In ACC on the top of page 34, I read: The construct Boo of boolean algebras is isomorphic to the construct BooRng of boolean rings and ring homomorphisms. This contradicts my intuition since ...
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1answer
47 views

Do surjective ring homomorphisms commute with intersection of ideals?

Let $f:A\longrightarrow B$ be a surjective ring homomorphism. Is it true that for any intersection of ideals, the image of the intersection is equal to the intersection of the images of the ideals? ...
2
votes
1answer
65 views

Nontrivial example of an artin algebra R such that R is pure-injective as an R-module

Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module. Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me ...
2
votes
1answer
50 views

Number of squares in $(\mathbb{Z}/p\mathbb{Z})^\times$

$x$ is a square in $(\mathbb{Z} /p \mathbb{Z})^\times$ iff there is a $y \in (\mathbb{Z} /p \mathbb{Z})^\times$ such that $x \equiv y^2 \mod p$. I am asked to show that there are exactly ...
0
votes
1answer
28 views

In a ring of characteristic 2 every prime ideal is maximal ideal

Let $R$ be a commutative ring with $1$ and $charR=2$.Then how can I show that every prime ideal in $R$ is a maximal ideal? I was trying to show it a boolean ring but I could not.Please Help ...
1
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1answer
26 views

Ring With Unity As a Direct Sum of Non-Zero Ideals

Let $R$ be a ring with unity. Let $ (a_i),i\in I $ be a family of non-zero ideals of $R$. Suppose that $R$ is a direct sum of the family $ (a_i),i\in I $ (i.e the additive group $R$ is a direct sum of ...
0
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1answer
68 views

Compute the units in $\Bbb Z_4[x]$.

Compute the units in $\mathbb{Z}_4[x]$. My Work: I have seen a problem to show the following statement (So, it is not a theorem): $R$-commutative ring with identity. Then $a_0+a_1x+\ldots ...
0
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1answer
151 views

Automorphism of an integral domain extends to an automorphism of the quotient field [closed]

Every automorphism of an integral domain can be extended to an automorphism of its quotient field. Please help to start with the proof!!
3
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0answers
57 views

Property of ideal [closed]

Let $R$ be an associative algebra. Let $I$ be an ideal of $R$. Let $J$ be an ideal of the algebra $I$. Prove that $(J)_R$ the ideal of $R$ generated by elements $J$, has the property: ...
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1answer
37 views

Problem on Integral Domain [duplicate]

Why $n\mathbb{Z} \times n\mathbb{Z}$ is not an integral domain?
4
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1answer
41 views

Left invertible matrices over rings with some special property

Suppose $R$ is a ring in which every left invertible element is invertible. Does this condition imply that every left invertible matrix in $\mathrm{M}_{n\times n}(R)$ is necessarily invertible?
3
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0answers
43 views

finite field extensions: how to compute norm and trace

I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is ...
3
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2answers
49 views

How to solve $xa=yb$ for $x,y$ in a ring

I'm probably missing something obvious, but does the equation $xa=yb$ necessarily have nontrivial solutions (where $x,y$ are not both zero) in a nonzero ring (i.e. $1\ne0$)? If one of them is zero, ...
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0answers
61 views

Reducing multivariate rational fractions to lowest terms

I wish to simplify multivariate rational fractions to a canonical form. Thanks to some very helpful mathematically inclined people who verified that my understanding of Wikipedia was correct, I'm now ...
1
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2answers
75 views

Solve $x^5 - x = 0$ mod $4$ and mod $5$

I'm trying to solve $$x^5-x=0$$ in $\mathbb{Z/5Z}$ and $\mathbb{Z/4Z}$ I don't see how to proceed, could you tell me how ? Thank you
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0answers
33 views

Using a Gauss sum to show that $p$ is of the form $x^2 + xy +3y^2$ if and only if $p \equiv 1, 3, 4, 5, 9 \pmod{11}$

Let $p \neq 11$ be an odd prime, and $\zeta$ an $11$th root of unity. Let $g$ be the Gauss sum $$g = \sum\limits_{i=1}^5 \zeta^{i^2} = \zeta + \zeta^4 + \zeta^9 + \zeta^5 + \zeta^3$$ We may ...
0
votes
1answer
40 views

Why is $I + (a) = R$?

Let $I$ be a maximal ideal of a ring $R$ and let $(a)$ be the principal ideal generated by the element $a$ which lies in $R$ but not $I$. Why does $R = I + (a)$? Thanks in advance
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1answer
32 views

showing an easy set is an ideal

I'm having troubles understanding the definition of an ideal. I found this example online, but the author did not explain the steps and just stated it was an ideal. I hope someone could show me the ...
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1answer
49 views

$f : A \to B$ be a ring homomorphism, projection map $ p: B \to B/M$, Is $p(f(A))/M$ isomorphic to $A/M$?

I am reading a book and I think it uses the following proposition Let $f:A \to B$ be a ring homomorphism where $A$ and $B$ are finitely generated $\mathbb{C}$ algebras and let $M$ be a maximal ...
0
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1answer
27 views

$R$ a conmutative ring, $K\subset R$ a field. ¿Does the characteristic of $\text{char}(R)=\text{char}(K)$?

Let $R$ a conmutative ring, $K\subset R$ a field. The characterist of $R$ is $n\in\mathbb{N} \:\mid\; \text{Ker}(\varphi)=(n)$, with $\varphi:\mathbb{Z}\longrightarrow R$ the canonical homomorphism. ...
2
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1answer
33 views

Prime and Maximal Ideals in Rings

Q. We know that if $R$ is commutative ring with unity, then (*) $M$ is maximal ideal $\Longleftrightarrow$ $R/M$ is field. (**) $M$ is prime ideal $\Longleftrightarrow$ $R/M$ is integral domain. ...
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1answer
32 views

local PID that is not a field is a DVR

I would be very happy if someone would check my proof of the fact that a local PID that is not a field is a DVR: Let $A$ be a local PID that is not a field. Since irreducibles generate maximal ideals ...
3
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1answer
45 views

$F \cong \mathbb Z_p$ for some prime p where $F$ is a field.

Suppose $F$ is a field and there is a ring homomorphism from $\mathbb Z$ onto $F$. Show that $F \cong \mathbb Z_p$ for some prime $p$. My try: If $F$ is of finite characteristics then it prime. So by ...
1
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1answer
95 views

$ \mathbb Z$ is not isomorphic to any proper subring of itself.

Show that the ring $ \mathbb Z$ is not isomorphic to any proper subring of itself. Is the cardinality main reason for not being isomorphic?? Please Help!!
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1answer
62 views

Show the quotient ring R/I is not a field

Studying for an exam in Algebra. Let $R=\mathbb{Z}[i]$ with the usual normfuction $N, z = 5+3i$ and $I = \, <z>$ Show that z isn't a prime element in $R$ and that $R/I$ isn't a field. I ...
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2answers
342 views

Is every prime element of a commutative ring “veryprime”?

Let $R$ denote a commutative ring. Define a function $$\| : R \times R \rightarrow \mathbb{N} \cup \{\infty\}$$ such that $a \| b$ is the number of times $a$ divides $b$ (and include $0$ in ...
2
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1answer
61 views

There are only two non isomorphic rings with $p$ elements

Prove that for any prime $p$ there are only two non isomorphic rings with $p$ elements. I have found out there are up to two rings of order p , they are $\mathbb Z_p$ and $\mathbb C_p$. Please ...
1
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1answer
34 views

$IJ$ is the set of nilpotent elements

Let $R$ be a commutative ring with identity which is Noetherian. Let $V(A)$ denote the set of all prime ideals of $R$ containing the ideal $A$. Suppose that $V(0) = V(I) \cup V(J)$ and $V(I) \cap V(J) ...
5
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1answer
78 views

Algebraic closure of the rational inside a quotient of product of finite fields

I'm trying to solve the following exercise: " Consider the ring $R = \prod_{p} \mathbb{F}_p$, where $p$ runs over all prime numbers and $\mathbb{F}_p$ is a field with $p$ elements. Show that there ...
1
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1answer
32 views

what inequalities can one have between $depth\ R$ and $depth\ M$? when $depth\ R \geq depth\ M$

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module. what inequalities can one have between $depth\ R$ and $depth\ M$? Obviously there are ...