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
22 views

Verifying an isomorphism between coordinate rings

I am trying to convince myself of an isomorphism between: $$k[x,y,z]/(x^2-yz,z-1) \rightarrow k[t]$$ In trying to show that these rings are isomorphic, I have constructed a map sending: $x ...
1
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2answers
212 views
+100

Show that the sequence is exact

We have that $R$ is a commutative ring. Suppose that $0\rightarrow A\rightarrow B\overset{f}{\rightarrow} C\rightarrow 0$ and $0\rightarrow C\overset{g}\rightarrow D\rightarrow E\rightarrow 0$ are ...
0
votes
2answers
19 views

Showing Quotient ring is a field using maximal Ideal

Question: Show that $R\left [ x \right ]/\left \langle x^{2}+1 \right \rangle$ is a field. Recall: Theorem: Let R be a commutative ring R with unity. Let I be a proper Ideal of a ring ...
0
votes
1answer
7 views

Verify size of factor ring

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$ and let I be the subset of R consisting of matrices with even ...
4
votes
1answer
40 views

Ring action on another ring?

So a module over a commutative ring $ R$ is an abelian group $G$ equipped with an action given by the product $R\times G\rightarrow G$ that satisfies a few conditions. What if $G$ itself is a ring? Is ...
0
votes
1answer
25 views

Kernel of ring homomorphism $k[X,Y] \rightarrow k[t^2, t^3]$ [duplicate]

Let $k$ be a field, $f: k[X,Y] \rightarrow k[t^2, t^3]$, $X \mapsto t^3, Y\mapsto t^2$. I would like to verify that $\ker f = (X^2 - Y^3)$. It is easy to see that $X^2 - Y^3 \in \ker f$ and ...
0
votes
0answers
28 views

What does $F[x]$ mean?

Lemma: $F$ is a field only if $F\left [ x \right ]$ is a Principal Ideal Domain. This is a theorem from Ring; divisibility of integral domain. What does $F\left [ x \right ]$ mean?
2
votes
1answer
21 views

Any ideal of a field $F$ is $0$ or $F$ itself

Prove that the only ideals of a field are $\left\{ 0 \right\}$ and the field itself. Let $F$ be a field and $I$ be an Ideal of $F$. Let $0 \ne x \in I$. Since $I$ is an Ideal of $F$, it is true ...
2
votes
1answer
46 views

Quotient of semi-simple rings

I can't seem to find a reference for the following statement: A quotient ring of a (left) semi-simple ring (left artinian with no nonzero nilpotent ideals) is a semi-simple ring. Does this ...
2
votes
1answer
42 views

Is there a way to characterize the prime ideals in $\mathbb{R}[x_1,x_2, \dots , x_n]$?

I'm studying algebras which can be formed by the quotient of principal ideals in $\mathbb{R}[x_1, \dots , x_n]$, and thus would like to be able to determine which of said principal ideals are maximal, ...
2
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2answers
20 views

Is a sub-algebra of a semisimple algebra nescesarily isomorphic to a subset of its direct product decomposition via Wedderburn's classification?

Wedderburn's classification of semisimple algebras tells us that any semisimple algebra $A$ is isomorphic to a finite direct product of matrix algebras over division algebras, say $A \cong ...
13
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1answer
379 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
-1
votes
1answer
19 views

On subring of a Field [on hold]

Let $R$ ba 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 are ideals of R, then either $I \subseteq J$ or $J \subseteq I$.
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0answers
7 views

A modularity condition

Let $R\subseteq S$ be rings with unity and $X$ ,$Y$ be subsets of $S$ with $X$ an ideal. If $S=X+Y$ what conditions should be held to infer the equality $R=(X∩R)+(Y∩R)$? I think that if we ...
1
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0answers
39 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: call $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b ...
1
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2answers
45 views

Prove that $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(i \sqrt{5})$ are not isomorphic.

The question is : Prove that $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(i \sqrt{5})$ are not isomorphic (I'm talking about ring isomorphism). What I have done : suppose there is an isomorphism ...
-3
votes
0answers
18 views

Prove that $M_p$ is a ideal of $\mathbb Z/(p)[x]$ and $\mathbb Z[x]/M$ is isomorphic to $\mathbb Z/(p)[x]/Mp$.

Let $M_p$ = $\gamma (M)$, the image of $M$ ($M$ is a maximal ideal of $\mathbb Z [x]$) in $(\mathbb Z/(p))[x]$, where $\gamma$: Z[x] --> Zp[x] is the morphism such that $\gamma (\sum_i a_ix^i)=\sum_i ...
1
vote
1answer
20 views

If $a \in A$ is not a prime number, then $A/aA$ is not an integral domain

If $a \in A$ is not a prime number, then $A/aA$ is not an integral domain: proof $a$ not prime, therefore: $a \mid bc$ and $a\nmid b$ and $a\nmid c$ Therefore $b+aA \neq aA$ and $c+aA \neq aA$ ...
1
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1answer
17 views

Showing an Ideal is the ring

If A is an Ideal of a ring R and the unity 1 belongs to A, prove that A=R. It is a sufficient condition to show that $A\subseteq R$ and $R\subseteq A$. Indeed, it is trivial to see that ...
1
vote
1answer
27 views

Intersection of any set of ideals is an ideal

Prove that the intersection of any set of Ideals of a ring is an Ideal. I'm looking for hints. Let A, B both be Ideals of a ring R. Suppose $I \equiv A\cap B$. Since A and B are both Ideals of ...
0
votes
1answer
16 views

Showing <n> =nZ is a prime ideal of Z IFF n is a prime

Question: If n is an integer greater than 1, show that $\left \langle n \right \rangle=n\mathbb{Z}$ is a prime ideal of $\mathbb{Z}$ IFF n is a prime. I have a bit of problem proving the ...
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0answers
26 views

Proof Verification of Result Involving Maximal Ideals

In further investigation of a question I asked earlier, I came across the following result, the proof of which I hope can be looked over here. I personally find it kind of interesting and I hope ...
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votes
1answer
49 views

How can we find the elementary divisors?

We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ($a\in R,x\in M$) is defined as usual if $a\in \mathbb{R}$, and $a\cdot x=(x_1, 0, 0)$ if ...
4
votes
2answers
168 views

Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$, prove that $\mathrm{Spec}(A_f)=\mathfrak{V}((f))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in ...
1
vote
3answers
26 views

Ideal generated by an element

Let $R\left [ x \right ]$ denote the set of all polynomials with real coefficients and let A denote the subset of all polynomials with constant term 0. Then A is an ideal of $R\left [ x ...
2
votes
2answers
66 views

Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
5
votes
2answers
103 views

Idempotent ideals in certain commutative rings

Let $R$ be a commutative ring with zero Jacobson radical such that each maximal ideal of $R$ is idempotent. Does it guarantee that each ideal is idempotent? I know only that if each maximal ideal ...
6
votes
1answer
87 views

What other properties follow from having a ring homomorphism to $\mathbb{Z}$?

(All my rings have $1$, and ring homomorphisms preserve $1$.) In $\mathbf{Set},$ the points of an object $X$ can be thought of as arrows from the terminal object $1$ to $X$. So I guess in general, we ...
1
vote
1answer
18 views

Visualising the kernel of a homomorphism and quotient groups

I am trying to represent myself quotient groups and I'm having trouble seeing what the kernel of a homomorphism : $\Phi: G \rightarrow G/H$ is (be it a ring homomorphism or a group homomorphism). I ...
0
votes
1answer
24 views

How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?

If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...
2
votes
3answers
53 views

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?

How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...
1
vote
1answer
34 views

Image of ring homomorphism is a subring

Let $\phi : R\to S $ be a ring homomorphism. Definition: $Im\left ( \phi \right )=\left ( R \right )\phi=\left \{ \left ( a \right )\phi \mid a \in R \right \}$ Prove that $Im\left ( ...
2
votes
1answer
306 views

Any ring is integral over the subring of invariants under a finite group action

I need to prove that if $G$ is a finite group that acts on ring $A$, and $A^G$ is the subring consisting of elements of $A$ which are invariant under all $g\in G$, then $A$ is integral over $A^G$. ...
2
votes
1answer
64 views

Prove that $R(+,.)$ is a division ring but I disproved it

QUESTION: Let $R=\left[\begin{matrix}\alpha & \beta \\ \bar\beta & \bar\alpha\end{matrix}\right]\in \mathbf{M_2(\mathbb{C})} $ where $\bar\alpha,\bar\beta$ denote the conjugates ...
-1
votes
1answer
32 views

Nilpotent matrix given nilpotent traces [on hold]

Let R be a conmutative ring and X a two by two matrix. Supose that Tr(X) and Tr(X^2) are nilpotent elements. Prove that 2X is nilpotent. Thanks a lot.
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0answers
17 views

concept of conjugacy class in a ring

Can we think of a similar concept of a conjugacy class in a ring which satisfies two three properties like conjugacy classes. I think of a set $M_x={xyx^{-1}-y}$ for $x\in R$ and $R$ is a division ...
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0answers
16 views

Proving that an element of a ring annihilates a module [duplicate]

Let $R$ be a commutative ring with $1$, $M$ be a finitely generated $R$-module, $\mathfrak{i}$ an ideal of $R$, and $\phi$ an $R$-homomorphism such that: $$1.\;\phi(M)\subseteq \mathfrak{i}M=M$$ i. ...
2
votes
2answers
40 views

on finite division subring of a ring

Is there any example of a ring which is not a division ring but any of its subring is a division ring? According to me if $R$ is a ring and $S$ is a division subring then $1\in S$ and hence ...
3
votes
0answers
43 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
0
votes
1answer
19 views

Splitting field and automorphisms

I know that if $K$ is a field and $f\in K[x]$, then there exists a splitting field of $f$ on $K$. If one has two isomorphic fields $K_1$ and $K_2$ (say $\sigma$ an isomorphism) and $f\in K_1[x]$, ...
2
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2answers
49 views

Irreducibility of polynomials in $\mathbf{Z}_p[x]$ - understanding proofs

I am reading through some irreducibility proofs and there's something I don't quite understand: $x^3+2x+1$ is irreducible in $\mathbf{Z}_3[x]:$ no roots in $\mathbf{Z}_3$ and degree $3$ so ...
-1
votes
0answers
34 views

$\mathbb Z[r_1,r_2,…,r_n] =\mathbb Z[\frac 1m]$ [duplicate]

Question Let $r_1,r_2, ...,r_n \in \mathbb Q $. Then $$\mathbb Z[r_1,r_2,...,r_n] =\mathbb Z[\frac 1m]$$ for some integer $m$. I think m must be the least common multiple of the ...
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votes
0answers
36 views

Application of generalized Chinese remainder theorem

Question Consider the ring $\mathbb Z[x]$ and define the ideals $I_p=(px-1)$ where p is prime Prove that $\mathbb Z[x]/I_2I_3...I_p$ is isomorphic to $\{\frac{n}{2^{a_2}3^{a_3}...p^{a_p}} | ...
1
vote
0answers
57 views

proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$

When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the ...
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0answers
41 views

How many elements are in the field of fractions $\Bbb Z_3(t)$?

As in exercise for my Galois Theory course I am supposed to find the number of elements in the field of fractions $\Bbb Z_3(t)$. I am very confused as to how to approach this question because I ...
1
vote
3answers
50 views

What is $\mathbb{Z_{n}}\left [ x \right ]$

Question: Show that $\mathbb{Z_{n}}\left [ x \right ]$ has characteristic $n$. What does $\mathbb{Z_{n}}\left [ x \right ]$ stands for? I'm very sure this is not the gaussian ring.
3
votes
3answers
47 views

What polynomial maps to $i$ under $\mathbb{Q}[x] \to \mathbb{Q}[x]/(x^2+1) \simeq \mathbb{Q}[i]$?

The rings $\mathbb{Q}[i]$ and $\mathbb{Q}[x]/(x^2+1)$ are isomorphic, and there is a surjective ring homomorphism from $\mathbb{Q}[x]$ to $\mathbb{Q}[x]/(x^2+1)$. Can someone give me an example of ...
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votes
2answers
23 views

How to prove the uniqueness of multiplicative identity?

Suppose $i_1, i_2 \in R$ which are multiplicative identity. Let $a$ also be in $R$. Then $a*i_1=a$ which means $a=i_2$. Thus, $i_2*i_1=i_2$. Now $a*i_2=a$, then $a=i_1$ hence $i_1*i_2=i_1$. Now how do ...
2
votes
1answer
29 views

How do I prove the uniqueness of additive identity?

First, suppose $i_1$ and $i_2$ are additive identity in ring R. From the definition of "additive identity" $a+i_1=a$ such that there is $a$ $\in$ $R$, including for $a=i_2$, so $i_2+i_1=i_2$. But ...
5
votes
3answers
4k views

Proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain

We're proving that $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain, using the norm function $$\nu (a + b\sqrt{2} ) = |a^2 - 2b^2|$$ and the first part says that since $\nu (a + b\sqrt{2} ) = |(a + ...