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Questions tagged [ring-theory]

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

-2
votes
1answer
25 views

Is $ \ f(x)=\ X^2 -22 \ X^2 +1$ Irreducible?

Is $ \ f(x) =\ X^4 -22 \ X^2 +1 \in \mathbb{Q[X]}$ The solution proved that it is irreducible, however it says that if f factores I've $\mathbb Z[X]$ then it is either can be factored to quadritic or ...
2
votes
1answer
35 views

Can there be other multiplications on $\mathbb{R}^2$ making it a ring?

Together with addition $+ : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$ $$(x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1 + y_2)$$ the multiplication $\cdot : \mathbb{R}^2 \times \mathbb{R}^2 \...
2
votes
2answers
26 views

Homomorphism in Rings

I have the following true/false statement from a worked example: If $F_1, F_2$ are fields and $\phi: F_1 \rightarrow F_2$ is a homomorphism, then $\phi$ is either the zero map or an isomorphism. ...
0
votes
2answers
35 views

Is there a noncommutative ring with quotient ring is commutative?

We have a result if R is commutative, then so is any quotient ring R/I, for any ideal I. If we take the contrapositive, we see that if R/I is non commutative, then R is non commutative. Is there any ...
0
votes
1answer
45 views

Error in “proof” that every quotient of PID is PID

I know that if $A$ is a PID and $I$ an ideal, then $A/I$ need not be a PID, since it's not even a domain unless $I$ is prime. However, I can't quite seem to find my mistake in the following "proof" ...
3
votes
3answers
45 views

$p^{n-1}x^n - 1$ over $\mathbb{Q}$ for $p$ prime

Consider $f(x) = p^{n-1}x^n - 1 \in \mathbb{Q}[x]$. I want to show that it's irreducible when $p$ is prime. Neither reduction of the coefficients modulo some prime nor Eisenstein seems to work here. ...
3
votes
1answer
57 views

Find all prime ideals of $\mathbb{Z} / n\mathbb{Z} \ [x]$

Let $n=p_1^{r_1}p_2^{r_2}\ldots p_k^{r_k}$ be an integer with $p_i$ being its prime factors. Find all prime ideals of $\mathbb{Z} / n\mathbb{Z} \ [x]$ containing monic polynomials of degree one. I ...
-2
votes
1answer
18 views

Is the set of all ideals of a quasi-local ring $R_P$ linearly ordered with respect to inclusion? [on hold]

Let $R$ be a commutative ring and $P\in\mathrm{Spec}(R)$. Let $R_P$ be the localization of $R$ at $P$. Is the set of all ideals of $R_P$ linearly ordered with respect to inclusion?
2
votes
2answers
22 views

Set of zero divisors and the set of units of the ring of functions from $X$ to $R$

Hope this isn't a duplicate. I was trying to answer the following question: Let $X$ be a non-empty set and $R$ be a ring. Then define $F(X,R)$ to be the ring of functions from $X$ to $R$. Then what ...
3
votes
1answer
98 views

Deciding whether the maximal ideal $\mathfrak{m}R_\mathfrak{m}$ is generated by two elements

Let $\phi$ be a ring homomorphism $\mathbb{C}[x,y,z] \to \mathbb{C}[s,t]$ such that $\phi(x) = s,\ \phi(y) = st,$ and $\phi(z) = t^2.$ Let $R = \operatorname{Im}\phi.$ Let $(a,b,c)\in\mathbb{C}^3$ and ...
-2
votes
2answers
39 views

descending chain of non-finitely generated ideals

Let $R$ be a commutative ring (not necessarily with unity) such that $ab\ne 0,\forall a,b\in R\setminus\{0\}$. If every descending chain of non-finitely generated ideals of $R$ terminate, then is it ...
1
vote
1answer
15 views

Is an $A$-module homomorphism of the $(A,B)$-bimodule $M$ also $B$-bimodule?

Let $A,B$ be rings, let $M$ be an $A$-module, which is, simultaneously an $A$-module and a $B$-module, and $a(xb)=(ax)b$ for all $a\in A$, $b\in B$, $x\in M$. Then for an $A$-module homomorphism of $...
1
vote
1answer
126 views

What's remarkable about this transfer of structure from $\Bbb Z[\frac12]$ to $\Bbb C$?

What, if anything, is remarkable about this transfer of structure from $\Bbb Z[\frac12]$ to $\Bbb C$? $Y=\Bbb Z[\frac12]\setminus0$ $X=\Bbb Z[\frac12]\cap(\frac12,1]$ $N=\{\ldots2,1,\frac12,\frac14,...
0
votes
0answers
49 views

Show that there is only one endomorphism of $\mathbb{C}[X,Y]/(X^2Y + XY^2 - XY)$ that is an automorphism on $\mathbb{C}$

Let $A = \mathbb{C}[X,Y]/(X^2Y + XY^2 - XY).$ Denote the images of $X,Y$ in $A$ as $\bar{X},\bar{Y}$ respectively. Let $\phi \colon A \to A$ be a ring homomorphism such that $\phi$ is an automorphism ...
2
votes
1answer
36 views

Find a simple formula of k numbers which gives output a number not among these k numbers.

This question just came to my mind while thinking. I just wanted to know if it is even possible. Anyway, the question goes as follows. You are given k numbers $a_{i}$ (k < n) from 1 to n. Your ...
0
votes
0answers
28 views

Direct proof for I prime ideal implies $I[x]$ prime ideal

I'm fully aware we can prove this by showing $I[x]$ is the kernel of the homomorphism from $R[x]$ to $(R/I) [x]$. but is the following proof valid? Suppose we have $p,q \in R[x],\not\in I[x]$ then ...
2
votes
2answers
16 views

Uniqueness Monic Polynomial Division Arbitrary Ring

Artin states "Let $R$ be a ring, $f$ be a monic polynomial and let $g$ be any polynomial, both with coefficients in $R$. There are uniquely determined polynomials $q$, $r$ in $R[x]$ such that $g = fq ...
0
votes
0answers
24 views

Application of forcing to first-order properties of rings

I'm not very well acquainted with forcing, just with the basic ideas; but I thought of the following proof, and since I'm definitely not comfortable with forcing I don't know if it's right, so my ...
-1
votes
0answers
10 views

TQuotient ring $frac{Z_2[x]}{<x^3-2,x-4>} $ [duplicate]

Please someone help me to list the elements the quotient ring as the following. Please.... $\frac{Z_2[x]}{<x^3−2,x−4>$
0
votes
1answer
37 views

Proving $(A + I) / I \cong A/(A \cap I)$

If $A$ is a subring of $R$ and $I$ is and ideal of $R$, let $A+I=\{a+i:a\in A,i\in I\}$, then prove $(A + I) / I \cong A/(A \cap I)$. We need the second theorem of homomorphisms but I need to find ...
-3
votes
0answers
29 views

The elements Quotient ring $\frac{Z_2[x]}{\langle x^3-2,x-4\rangle } $ [on hold]

Please someone help me to list the elements of the quotient ring as the following. Please... $\frac{Z_2[x]}{\langle x^3-2,x-4\rangle} $
1
vote
1answer
14 views

On the two definitions of Rings of Quotients in T.Y.Lam

In the book "Lectures on Modules and Rings" by T.Y.Lam there are two definitions: ((8.2) Definition, T.Y.Lam.) We say that $N$ is a dense submodule of $M$ (written as $N\subseteq_d M$) if, for any $...
0
votes
1answer
36 views

Gauss' Lemma prove $\mathbb{Z}[x]$ UFD

I am trying to deduce that $\mathbb{Z}[x]$ is a UFD given the fact that the product of two primitive polynomials $fg$, given $f,g\in{\mathbb{Z}[x]}$, is primitive (I have managed to prove this myself)....
1
vote
1answer
65 views
+200

Short proof of $\mathbb{Q}[x,y]/\langle x^2+1, y^4-2\rangle \equiv\mathbb{Q}[\sqrt[4]{2}, i]$

I am looking for an indirect proof of $$E = \mathbb{Q}[x,y]/_{\langle x^2+1, y^4-2\rangle}\cong\mathbb{Q}[\sqrt[4]{2}, i],$$ much preferably using module homomorphism theorems. To be more specific, ...
2
votes
1answer
29 views

For a finite commutative ring $R$, $a \in R$ is a root of $p(x)$ iff $p(x)$ can be written as $p(x) = (x-a)g(x)$

$R$ is a finite commutative ring with identity. Let $p(x) \in R[x]$, the ring of polynomials over $R$. Show that $a \in R$ is a root of $p(x)$ if and only if $p(x)$ can be written as $p(x) = (x-a)g(...
0
votes
0answers
26 views

On contraction of maximal ideals in simple integral extension

Let $R\subseteq T$ be integral domains. Let $a\in T$ be a root of some degree $d$ monic polynomial in $R[X]$. Then of-course $R[a]$ integral over $R$. I am trying to show, without using any heavy ...
4
votes
2answers
65 views

Dummit and Foote problem 11 in section 7.4

I am trying to solve problem 11 in Dummit and Foote section 7.4. The problem is the following: Assume $R$ is commutative. Let $I$ and $J$ be ideals of $R$ and assume $P$ is a prime ideal of $R$ that ...
0
votes
0answers
12 views

Given $P_1,P_2\in \mathbb R^2$, integer $n>1$, and $f\in \mathbb R[X,Y]$, $f=g_1+g_2$, partial derivatives of order $<n$ of $g_i$ vanishing at $P_i$

Let $P_1,P_2\in \mathbb R^2$ and let $n\ge 1$ be an integer. Given $f(X,Y)\in \mathbb R[X,Y]$, how to show that there exists $g_1(X,Y),g_2(X,Y)\in \mathbb R[X,Y]$ such that $f(X,Y)=g_1(X,Y)+g_2(X,Y)$ ...
0
votes
2answers
36 views

Explain why the ideal $(x,y)$ of $\mathbb Q[x,y]$ is not generated by a single element.

Explain why the ideal $(x,y)$ of $\mathbb Q[x,y]$ is not generated by a single element. I know the ideal $(x,y)$ is maximal since $\mathbb Q[x,y]/(x,y) \cong \mathbb Q$; and I know $\mathbb Q[x,y]$ ...
0
votes
1answer
32 views
+50

What is the meaning of presentation of an unital associative Ring?

Let $R$ be an unital associative ring and let $f: F \rightarrow R$ be an onto ring homomorphism. Where F is some freely generated ring over the set $S$ then $<S|T>$ is called presentation of ...
1
vote
4answers
46 views

Show that in the ring $R = \mathbb{Q} [x, y]$ there are ideals that require at least two generators (the ideal $I =\{f\in R: f (0,0) = 0\}$)

Show that in the ring $R = \mathbb{Q} [x, y]$ there are ideals that require at least two generators (for example, the ideal $I =\{f\in R: f (0,0) = 0\}$) What would be the generators for the example ...
1
vote
0answers
394 views

Coefficients such that linear combination lies in an ideal

Let $R$ be a ring, $I$ an ideal, and $\langle g_1, \ldots, g_m \rangle$ a finitely generated ideal. Considering the intersection $I \cap \langle g_1, \ldots, g_m \rangle$, I became interested in the ...
1
vote
2answers
18 views

Defining multiplication of quotient ring with respect to polynomial

Let $R=\mathbb{Z}_{11}[x]/(x^2+2)$. I would like to define multiplication in $R$. To my understanding, it is required that if $[ax+b][cx+d]=[rx+s]$ we want to find $r$ and $s$ in terms of $a,b,c,d$. ...
0
votes
1answer
14 views

Find number of polynomials in $P_n(F)$ where $F$ is a finite field with cardinality $m$. [duplicate]

Firstly, I should mention that the above question hits my mind when I am solving a problem in Abstract Algebra which is- Find the number of elements in $\frac{\Bbb{Z}_3[x]}{<x^3+2x+1>}$. I have ...
3
votes
1answer
53 views

Homomorphism from $\mathbb R^2\to \mathbb C$

Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows: $(a,b)(c,d)=(ac,bd)$
3
votes
1answer
42 views

Non-unital ring contains a unital ring as a subring?

It is well known that every non-unital ring can be embedded into a unital ring (e.g., Dorroh's adjunction). I am curious about the converse: every unital ring can be viewed as a subring of a non-...
1
vote
2answers
24 views

Ideal quotients - when does $I:h^2 = I:h$ hold?

A professor gave me the following problem: prove the fact that $I : h^2 = I:h$, where $I \subset k[x_1,\dots,x_n]$ is a zero-dimensional ideal, and $h$ has the property $I + (h) = I + (h^2)$. Now I ...
0
votes
2answers
30 views

How to multiply in the formal Laurent series ring

I'm working on a problem that asks me to show that the ring of formal Laurent series $F((x))$ is actually a field when $F$ is a field. My problem is that the problem doesn't define the product between ...
5
votes
0answers
72 views

Finding an explicit map associated with the quotient ring of Gaussian integers [duplicate]

Let $a+bi\in\mathbb{Z}[i]$ with $\gcd(a,b)=1$. I know that $\mathbb{Z}[i]/\langle a+bi\rangle\cong\mathbb{Z}_{a^{2}+b^{2}}$ by a ring homomorphism $\phi:\mathbb{Z}[i]\to\mathbb{Z}_{a^{2}+b^{2}}$, ...
10
votes
2answers
408 views

Every finite commutative ring with no zero divisors contains a multiplicative identity?

There is an easy argument which shows that a finite integral domain (commutative unital ring with no zero divisors) is a field. Here I wonder whether this result still stands if the term "unital" is ...
1
vote
1answer
30 views

What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field?

What does it mean for a member of formal power series over a field to be algebraic over polynomial ring of that field? For example what does it mean for a $f$ in $k[[t_1 ,...,t_n ]]$ which is ...
0
votes
1answer
31 views

Unital rings correspondence by a ring homomorphism

Let $\phi:R\rightarrow R'$ be a (not necessarily unital) ring homomorphism. If $S'$ is a subring of $R'$, it is easy to show that $\phi^{-1}(S')$ is a subring of $R$. But if $S'$ is also unital, will $...
1
vote
1answer
32 views

Cardinality of generated rings and generated modules

I've once asked a similar question only about groups, but I am interested whether the logic is still sound: $(1)$Let $S$ be a generating set of a ring $R$, and denote $\kappa=\vert S\vert$. Then $\...
1
vote
1answer
23 views

Ring homomorphism between unital rings

Let $R,R'$ be unital rings. Let $\phi:R\rightarrow R'$ be a homomorphism such that $1_{R'}\in{\sf im}\phi$. Does this suffice to say $\phi(1_R)=1_{R'}$? Or does there exist any counterexample? PS: ...
0
votes
2answers
49 views

Prove that there is a nontrivial ring homomorphism from $\mathbb{Q}[x,y]/(x,y)^2$ to $\mathbb{C}$.

I am currently studying for my qualifying exam in algebra. I am thinking about an old question that asks you to look at $R = \mathbb{Q}[x,y]/I$ where $I=(x,y)^2$. It asks for the following three ...
1
vote
1answer
68 views

Does there exist a polynomial it does not have any integer root but has at least one root in $\Bbb Z_n$ , $\forall n \in \Bbb Z$?

Does there exist a polynomial $f \in \Bbb Z[X]$ such that $f$ does not have any integer root but $f$ has at least one root in $\Bbb Z_n$ , $\forall n \in \Bbb Z$ ( while considering $f$ as a ...
1
vote
1answer
48 views

Noncommutative rings and prime/maximal ideals

Let $R$ a non-simple noncommutative ring, and let $\mathcal{I}$ the set of non-trivial (right, left) ideals of $R$, with the following property: "Every element $I \in \mathcal{I}$ is prime and/or ...
-1
votes
0answers
25 views

Does the action of the free abelian group $\Bbb Z^2$ generalise to $\Bbb Z^{n+1}$ on $\Bbb Z\left[\frac1{\prod_n p_n}\right]\setminus0$?

Does the action of the free abelian group $\Bbb Z^2$ generalise to unlimited many dimensions: $\Bbb Z^{n+1}$ on $\Bbb Z\left[\dfrac1{\prod_n p_n}\right]\setminus0$? First consider the following ...
1
vote
0answers
30 views

Example of commutative ring with two elements that don't generate entire ring

I believe that there should be an example of a commutative ring $R$ that contains two elements whose only common divisors are units but which do not generate the unit ideal. $R$ can't be a Euclidean ...
1
vote
1answer
31 views

Localization of a finitely generated module is nonzero iff the anihilator is contained in the prime ideal?

I'm a bit stuck on both directions. I think I need to use the fact that M is finitely generated and maybe find a way to show each of the generators is killed by everything and that would imply $M_p$ ...