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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5
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1answer
49 views

Is the ring of Laurent polynomials in $n$ noncommuting variables Noetherian?

Suppose we have a Noetherian ring $R$. Is it true that the ring of Laurent polynomials $R\langle x_1,\,x_1^{-1},\ldots,\,x_n,\,x_n^{-1}\rangle$ in $n$ noncommuting variables is also Noetherian? If so, ...
0
votes
3answers
47 views

How to determine non trivial homomorphisms [closed]

I am trying to understand and it doesn't make any sense to me: How can I determine if there are any non trivial homomorphisms between groups or rings? How do I find them? and once I found them, how ...
1
vote
0answers
56 views

searching about an isomorphism

I'm looking for an isomorphism : $$H: \overbrace{\mathbb{F}_q^r\oplus\cdots\oplus \mathbb{F}_q^r}^{l\text{ times}}\longrightarrow \frac{\mathbb{F}_q[X_1,\ldots,X_l]}{(X_1^r-1)\cdots(X_l^r-1)}$$
0
votes
0answers
23 views

Noether & Schmeidler- Hurwitz-Ideals

Consider the following page from Noether and Schmeidler's 1921 work: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0008&DMDID=DMDLOG_0008&LOGID=LOG_0008&PHYSID=PHYS_0013 ...
4
votes
2answers
68 views

Dimension of the affine variety associated to $\langle zw-y^2, xy-z^3 \rangle $

Find the dimension of the affine variety $V(I)$, where $I=\left\langle zw-y^2,xy-z^3\right\rangle \subseteq k[x,y,z,w]$, with $k$ algebraicaly closed field. I tried to solve the system $zw-y^2=0$, ...
2
votes
1answer
56 views

What does $(G:G')$ mean?

I'm trying to teach myself some ring theory from a book, and have come across this sentence: "There are $(G:G') = 4$ linear characters" where $G$ is a group, and $G'$ is the derived group. I ...
1
vote
2answers
59 views

Prove that field $Q(x)$ is a field of fractions of ring $F[x]$

Let $F$ be a commutative ring without zero divisors and $Q$ its field of fractions. How can I prove that field $Q(x)$ is a field of fractions of ring $F[x]$? And also why is it that field $Q((x))$ ...
1
vote
0answers
33 views

Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
2
votes
1answer
32 views

Show that $G$ is a basis of group ring $RG$ over $R$.

Show that $G$ is a basis of group ring $RG$ over $R$. Comments: That $G$ is through direct generator, if $\alpha \in RG$ then $\alpha = \sum_{g \in G} a_gg$. I am not able to show that $RG$ is a ...
1
vote
0answers
37 views

Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left ...
0
votes
1answer
40 views

Please give an example of a ring that does NOT have a multiplicative identity but contains a subring that does have an identity..

I cannot think of an example of such a ring. For that matter, other than the even integers I cannot even think of an example of a ring without an identity.
13
votes
5answers
802 views

Pathologies in “rng”

There is no general consensus regarding whether a ring should have a unity element or not. Many authors work with unital rings , and other does not essentially require unity. If we do not assume ...
5
votes
0answers
23 views

Why not take the tensor product of two left modules in this way? [duplicate]

Let $A,B$ be two left $R$-modules. I was wondering if we then can form the tensor product of $A$ and $B$ by the free abelian group on $A \times B$ divided out by the span of the following elements ...
1
vote
1answer
38 views

How does one find the Krull dimension of a composite ring?

For example, if the ring is $\mathbb{Z} + X \mathbb{Q}[X]$. Is the dimension $1$?
0
votes
0answers
50 views

What are the ring theoretic properties of this trivial extension? [closed]

Let $R={\mathbb Z }_{ (p) }{ \ltimes\mathbb Z }_{ { p }^{ \infty } }$ be the trivial extension of ${\mathbb Z }_{ (p) }=\{ a/b\in\mathbb Q:b\neq 0,\gcd(a,b)=1\text{ and } p\nmid b\}$ by ${\mathbb Z ...
1
vote
1answer
33 views

Why is $(2,x)$ non-principal in $\Bbb Z[x]$? [duplicate]

Why is $(2,x)$ non-principal in $\Bbb Z[x]$? Apparently this is the case, I just read it on wiki, as a counter example to $\Bbb Z[x]$ being a PID. What is $x$ here? I mean $2$ can surely generate ...
2
votes
0answers
39 views

Ring structure of a localization

let $R$ be a commutative noetherian ring and let $A$ be an $R$-algebra which is moreover a finitely generated $R$-module. Let $P$ be a prime ideal of $R$. How is the ring structure of the localization ...
0
votes
1answer
34 views

Finding all homomorphisms between rings

I am looking for a good method to understand how to find all possible homomorphisms between rings, e.g $\varphi :\mathbb{Z}\rightarrow \mathbb{Z}$ or, as another example: $\varphi ...
1
vote
1answer
37 views

Find all subrings of a ring

Given a finite ring, e.g $\mathbb{Z}{_{24}}$, how can I find all of its subrings? I have tried to think about it couldn't reach any idea. Thanks.
1
vote
1answer
81 views

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
0
votes
1answer
14 views

Prove $\operatorname{ann}_r(S)$ is an ideal.

Let $S$ be a right ideal of ring $R$. Let $\operatorname{ann}_r(S) = \{r \in R:ar=0 \mbox{ for all } a \in S\}$. Prove $\operatorname{ann}_r(S)$ is an ideal of $R$. I have practically no experience ...
3
votes
2answers
264 views

Example: Krull dimension 1 but not a PID

It's easy to prove that if $A$ is a PID which is not a field then $\dim A= 1$. What is a counterexample to the converse? Thanks for any insight.
1
vote
2answers
28 views

Invertibility of translating ideals from a ring to its localization

Let $A$ be a ring with some multiplicative subset $S$. Define $AS^{-1} = \{\frac as| a \in A, s \in S\}$. Let $I_A$ and $I_S$ be the sets of ideals of $A \subset A-S$ and $AS^{-1}$ respectively. ...
0
votes
1answer
67 views

Finding the quotient ring $\mathbb{Z}[i]/(4+i)$

Find the quotient ring $\mathbb{Z}[i]/(4+i)$ by identifying elements with the lattice points in the square generated by $4+i$. I know that $N(4+i) = 17$. Therefore, $4+i$ is irreducible. Now ...
1
vote
0answers
16 views

Computing injective hulls over a lower triangular matrix ring

Let $R$ be the ring $\begin{pmatrix} {\mathbb Z }_{ p } & 0 \\ {\mathbb Z }_{ p } & {\mathbb Z }_{ p } \end{pmatrix}$ with $p$ prime. $R$ is an artinian ring and is also a $\mathbb ...
0
votes
3answers
47 views

Proof of ring isomorphism

Proof that $Z[X]/(X^2-22) ≈ Z[\sqrt{22}]$. I have tried all sorts of things to resolve this but I don't know how to wrap my head around it. Can you please explain how to solve these kind of ...
0
votes
1answer
22 views

$E_{i,j} = E_{i,r} * E_{r,s} * E_{s,j} (i,j=1,2,…,n)$ where $E_{x,y}$ is an $n \times n$ matrix with a 1 in row x column y and 0's otherwise.

This matrix equation was given in my ring theory text as a step in showing that there are no nontrivial proper ideals of the ring of $n \times n$ matrices with real number entries. I am assuming it ...
1
vote
2answers
46 views

Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
1
vote
1answer
29 views

Quotient ring is a ring homomorphism

Why is this a ring homomorphism? $$\phi:R\to R/I$$ where $I$ is an ideal, given by $\phi:r\mapsto r+I$. To be a ring homomorphism it needs to be a homomorphism of addition and multiplication, i.e: ...
3
votes
1answer
52 views

Are these subrings of $\Bbb Q$?

Are the following subrings of $\Bbb Q$? 1) The set of non-negative rational numbers. No since we don't have any additive inverses, and the subring should be armed with an Abelian group for ...
0
votes
3answers
31 views

Prove for every $a$ in $I$ and every $b$ in $J$ that $ab=0$.

$I$ and $J$ are respectively right and left ideals of ring $R$. $I$ and $J$ have no elements in common other than $0$. Prove for every $a$ in $I$ and every $b$ in $J$ that $ab=0$. I have ...
4
votes
2answers
42 views

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring)

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring) : $ \triangleleft $ means ideal. I need this proof to continue on, I'm told it's not that hard, but I just ...
1
vote
2answers
72 views

Converse of Hilbert's Nullstellensatz [duplicate]

I'm referring to the following version of the Nullstellensatz: If $k$ is algebraically closed, then every maximal ideal of $k[X_1,\ldots,X_n]$ is of the form $(X_1-\lambda_1,\ldots,X_n-\lambda_n)$ ...
0
votes
0answers
45 views

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$ My attempt: Note that $\Bbb F[x]$ where $\Bbb F$ is any field is a Euclidean domain, and importantly, that means that ...
4
votes
3answers
32 views

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?(When the fraction is completely reduced) I have tried to apply the subring test on this, and this means I want to show ...
5
votes
2answers
101 views

Existence of ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field

Does a ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field with characteristic $p\equiv 3 \bmod 4$ such that the unity is mapped onto the unity exist? Thank for your help.
3
votes
0answers
63 views

In the ring of polynomials $F[t]$, every ideal is principal

Let $$\langle a \rangle=\bigcap_{ a \in I} I$$ $$ \langle a \rangle =\{ Ira + na, r\in R,n \in Z\}$$ What is unclear is why$$ I=\langle 0 \rangle$$ proves that I is a principal ideal. My definition ...
2
votes
1answer
50 views

Quotients of the ring of Laurent polynomials in one variable

I am trying to understand quotients of the ring $R=k[X,X^{-1}]$, where $k$ is a finite field. I note that $R$ is a PID since it is the localization of a PID; namely $k[X]$ localized at ...
2
votes
2answers
54 views

Naïve groups, fields and ideals

Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right ...
3
votes
2answers
58 views

Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
0
votes
3answers
26 views

Prove $a\mathbb{Z}[x]+x\mathbb{Z}[x]$ is a principal ideal on $\mathbb{Z}[x] \iff a=0$ or $a=1$ or $-1$.

I've tried several things, but I don't know how to properly show it. Prove $a\mathbb{Z}[x]+x\mathbb{Z}[x]$ is a principal ideal on $\mathbb{Z}[x] \iff a=0$, $a=1$ or $a=-1$. My try: Proof: ...
1
vote
1answer
42 views

Proving that structure is a ring.

Lets say, I have a set S with two operations defined in it: + and *. I need to prove that this structure is a ring. I also have a structure R, which i know is a ring, and a function: $$f: S ...
2
votes
1answer
41 views

Ring localization and ideals

I'm trying to solve a couple of problems involving ring localization and I'm not sure if my solutions are right or if I understand the idea of localization correctly. Let $A$ be a commutative ...
1
vote
1answer
39 views

Descending chain of ideals becoming stationary

Exercise 3.7 of Algebraic Number Theory (Neukrich) is: In a noetherian ring R in which every prime ideal is maximal, each descending chian of ideals $\mathfrak{a_1 \supset a_2 \dots}$ becomes ...
1
vote
0answers
35 views

Generalized fact in ring theory about irreducible elements

It is quite easy to show that for $A$ an integral domain, an element $a \in A$ is irreducible if and only if the principal ideal $\langle a \rangle$ is maximal for inclusion among proper principal ...
2
votes
0answers
42 views

Idempotent semiring

Let $R$ be a semiring. For $a\in R$,we define $t_a(x)=x+a$ for all $x\in R$. Prove that $R$ is idempotent(with +) and $1$ has an infinite order if and only if for all $a,x,y\in R$, ...
1
vote
1answer
54 views

The existence of a polynomial factor

Given two polynomials $p_1(x_1,\dots, x_m)$ and $p_2(x_1,\dots, x_n)$ over reals, where $m > n$, and we know that $p_2(x_1,\dots, x_n)=0 \implies p_1(x_1,\dots, x_m) =0$. My question is: ...
1
vote
1answer
31 views

Let $\Phi : R \rightarrow R'$ be a ring homomorphism, where $R,R'$ are rings with unity. Then which of these is true?

Let $\Phi : R \rightarrow R'$ be a ring homomorphism, where $R,R'$ are rings with unity. Then which of these is true : $(i)~\Phi(1)=1 ~\forall~$ rings $R,R~'$ with unity $(ii)~\Phi(1) \ne 1 $ for ...
2
votes
1answer
49 views

Principal ideal domain, $\forall x=(x_1,x_2)^t \in R^2~~\exists G \in SL_2(R) : Gx=(\gcd(x_1,x_2), 0)^t$

Let $R$ be a principal ideal domain. Prove that for every $x=(x_1,x_2)^t \in R^2$ exists a matrix $G \in SL_2(R)$ for which $Gx=(\gcd(x_1,x_2), 0)^t$. I think it's easy, but do not know how to ...
0
votes
2answers
83 views

Can We Always Build a Field out of an Integral Domain?

Link to Hungerford's Text Let $R$ be an integral domain, and $F$ its quotient field (or field of fractions). Assuming that $\phi: R \rightarrow F$ is isomorphic, $R[x]$ is isomorphic with $F[x]$ ...