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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4
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1answer
21 views

Module isomorphism from $R$ to $R \oplus R$ for a certain ring $R$

My textbook says: Let $R$ denote the set of infinite–by–infinite, row– and column–finite matrices with complex entries. Show that $R \cong R \oplus R$ as $R$–modules. So for $A, B \in R$, I tried ...
7
votes
2answers
163 views

What's the theoretical basis for integration using partial fractions?

Exercises involving integration using partial fractions depend on expressing a rational function $\frac{P(x)}{Q(x)}$ (where the degree of $P$ is less than the degree of $Q$) as a sum of ...
0
votes
0answers
32 views

How to test if a Subset of a Ring is an Ideal.

I've been browsing Ideal test and I'm trying to understand what constitutes a concise and efficient method of testing to see if a subset is an ideal of a ring. I understand For $I$ to be an ideal of ...
2
votes
0answers
34 views

Euclidean domains and Fields

I've been wrtiting a chain of inclusions of algebraic structures as given at the end of this first paragraph on wikipedia: http://en.wikipedia.org/wiki/Euclidean_domain And I've been giving examples ...
1
vote
2answers
43 views

What are the semisimple $\mathbb{Z}$-modules?

What are the semisimple $\mathbb{Z}$-modules? Comments: I think they are direct sums of copies of such $\mathbb{Z}_p$'s, where $p$ is a prime number. I believe it is, but I can not prove.
3
votes
3answers
48 views

Integral Domains and Unique Factorisation Domains

I'm learning about Rings, commutative rings, IDs, UFDs, etc with each being a subset of the predecessor, and I'm now trying to find an ID that is not a UFD I understand $\mathbb Z[\sqrt{-5}]$ is an ...
3
votes
1answer
37 views

Ring automorphisms of $\mathbb{Q}[\sqrt{2}]$

What are the all possible ring automorphisms of $\mathbb{Q}[\sqrt{2}]$? According to me, it is completely determined by its value on $\sqrt{2}$. Am I correct?
0
votes
0answers
14 views

Let P be a proper left ideal of R. Want to show that if P is comaximal with every non zero 2 sided ideal of R, Core(P) = {0}.

Let P be a proper left ideal of R. Want to show that if P is co-maximal with every non zero 2 sided ideal of R, Core(P) = {0}. The definition I am using of comaximal is: "I is comaximal with J if ...
2
votes
1answer
38 views

How to show that $\mathbb Z+x \mathbb Q[x]$ is a GCD domain ?

How to show that $\mathbb Z+x \mathbb Q[x]$ is a Bezout domain , that is sum of two principal ideals is again a principal ideal ? Or at least , how to show that it is a GCD domain ? ( This will then ...
1
vote
1answer
33 views

Looking for an example of a GCD domain which is not UFD

I know that every UFD (unique factorization domain ) is a GCD domain i.e. g.c.d. of any two elements , not both zero , exists in the domain . I am looking for an example of a GCD domain which is not ...
1
vote
2answers
43 views

Prime Ideals and multiplicative sets

I am currently studying a course on commutative algebra and came across this statement: An Ideal $I$ in a ring $R$ is prime if and only if $R\setminus I$ is a multiplicative set. I have proved ...
1
vote
3answers
86 views

Show that $R/(I \cap J) \cong (R/I) \times (R/J) $

My question actually follows from this one: Show that if $I + J = R$, then $R/(I \cap J) \cong R/I \times R/J$ What I don't understand is why is it necessary for $I+J=R$, in order for $$ ...
1
vote
2answers
30 views

Why is the Jacobson radical of the integers {0}.

Why is the Jacobson radical of the integers {0}? I have been working through questions dealing with the Jacobson radical and have come across this and can't think of why this would be. Any help ...
2
votes
1answer
85 views

When is $G\cong\operatorname{End}(G)$?

$\newcommand\End{\operatorname{End}}$Let $G$ be an Abelian group. Are there sufficient conditions for the existence of an isomorphism $G\cong\End(G)$, where $\End(G)$ is considered a group under ...
0
votes
1answer
52 views

The relation between prime ideal and simple ring

I saw "Any simple ring is a prime ring" as an example in Prime ring@wiki. Can anyone show me how to proof it? Also, on the other side, is any prime ring a simple ring? Thanks.
1
vote
4answers
72 views

All prime ideals are maximal - Counterexample

I would like to know of some simple counter examples to the statement "ALL prime ideals are maximal" I say counter examples because I think the statement isn't true.
2
votes
4answers
159 views

$x$ is a left zero-divisor $\iff$ $x$ is a right zero-divisor.

Let $R$ be a ring with unity. Show that $x$ is a left zero-divisor if and only if $x$ is a right zero-divisor. Suppose, $x$ is a left zero divisor. Then, $\exists y \in R$ such that $xy = 0 ...
3
votes
1answer
34 views

Further examples of Principal Ideal Domain that are not Euclidean Domains

In several courses of algebra, I've heard that not all PIDs are EDs, and the canonical example is $\mathbb{Z}\left[\dfrac{1+\sqrt{-19}}{2}\right]$ which I've heard over and over. Some cursory research ...
1
vote
1answer
35 views

Show that if $M$ is a semisimple artinian module then $M$ is finitely generated.

The exercise is as follows: Show that for a semisimple module $M$ over any ring, the following conditions are equivalent: $(1)$ $M$ is finitely generated; $(2)$ $M$ is Noetherian; $(3)$ $M$ is ...
1
vote
0answers
24 views

Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Can one compute $\ell_S(S/(a_1^i,a_2^j))$?

Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Let $x_i$ be the natural image of $X_i$ in $S$. Set $a_1=x_1+x_3$ and $a_2=x_2+x_3$. $a_1,a_2$ is a system of parameters of $S$. So ...
0
votes
2answers
44 views

Question about ideals

I believe the following is a true statement, but I am unsure, so I wanted to check with people. If $p$ is an irreducible polynomial in $n$ indeterminates then $(p)$, the ideal generated by it, is ...
0
votes
1answer
27 views

Proving $f(x)=x^n-p$ is minimal of $\alpha=\sqrt[n]{p}$ over the field F (p is prime)

I am having trouble with the concept of minimal polynomial, In a homework question I have concluded the following: $\mathbb{Q} \le \mathbb{F} \le \mathbb{C}$ - field extensions such that ...
1
vote
2answers
52 views

Irreducible polynomial over Q

Let $f(x) = 3x^4+6x^3+24x^2+18 \in \mathbb Z[x]$. Is $f(x)$ irreducible over $\mathbb Q$ ? In my course, Eisenstein's criterion is apply for monic polynomial only, hence, I can't use it with p =2. If ...
0
votes
1answer
37 views

Why are ring extensions only discussed in the context of $\mathbb{C}$?

I'm watching the great (imo) set of lectures on abstract algebra from the harvard extension school that's available on youtube. Now this lecture is about extending a ring. The lecturer talk about ...
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vote
0answers
13 views

Prove the annihilator of a quotient module is a 2 sided ideal. [duplicate]

R is a ring and is a left ideal of R. Since R is a ring, it can also be considered as an R-module, therefore I is a submodule of R and the quotient R/I is an R-module. I am trying to solve the ...
1
vote
1answer
46 views

In every ring with unity satisfying ACC, every ideal is finitely generated; can we prove it without assuming Axiom of choice?

Assuming Zorn's lemma, we can prove that in every ring with unity satisfying Ascending-Chain-Condition, every ideal is finitely generated. Is this statement equivalent to Zorn's lemma? Can we prove it ...
1
vote
1answer
63 views

In any commutative ring with unity, every prime ideal is finitely generated implies every ideal is finitely generated; can it be prove without A.C.?

Assuming Zorn's lemma, "In any commutative ring with unity, if every prime ideal is finitely generated, then every ideal is finitely generated". Is the converse true, i.e. if in any commutative ring ...
1
vote
1answer
40 views

If $I$ proper ideal of $R$, $S$ ring extension of $R$, and $u$ a unit in $S$, then $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$

Let $R ⊆ S$ be an extension of rings, and let $u$ be a unit in $S$. Let $I$ be an ideal of $R$ with $I \ne R$. Show that $IR[u] \ne R[u]$ or $IR[u^{−1}] \ne R[u^{-1}]$. Here is what I try: I have ...
0
votes
1answer
18 views

gcd in principal ideal domain

Let R be a principal ideal domain. Show that any pair of nonzero elements a, b in R have a greatest common divisor and that for any greatest common divisor d, we have d in aR + bR. Show that a, b are ...
1
vote
1answer
28 views

List all homomorphisms from $\mathbb{Z_2}$ to $\mathbb{Z_4}$

I have only come up with two. That is $f(1)=1, f(1)=0$. Are there any more? If so, how should I go about thinking this problem through, to ensure that I have found all of them?
1
vote
1answer
32 views

Let $D$ a division ring. Show that $Z(M_n(D)) \simeq Z(D)$

Let $D$ a division ring. Show that $Z(M_n(D)) \simeq Z(D)$. I'm having trouble mounting the isomorphism, do not know how to proceed.
1
vote
1answer
54 views

Find $\alpha$ such that the given field is $\mathbb{Q}(\alpha)$ [duplicate]

This question is in regards to separable field extensions. I am to show that this $\alpha$ is in the given field and verify by direct computation that the given generators for the extension of ...
0
votes
1answer
29 views

Let $I,J,K$ ideals of ring R. Prove that $I+JK \subseteq (I+J)(I+K)$

Let $I,J,K$ ideals of ring R. Prove that $I+JK \subseteq (I+J)(I+K)$ Comments: This is something I need to solve an exercise. Previously tasted the ring I'm working $IJ = I \cap J$ should the need ...
1
vote
0answers
24 views

prove n divides $[\mathbb{F}[\alpha]:\mathbb{Q}]$

$\mathbb{Q}<\mathbb{F}<\mathbb{C}$ - field extensions, such that $[\mathbb{F}:\mathbb{Q}]=m \in \mathbb{N}$ p is a prime number, $\alpha=p^{\frac{1}{n}}$ gcd(m,n)=1 prove n divides ...
1
vote
2answers
45 views

Quick question on $x^{3}-2$ and field extensions.

For the case where $[E:\mathbb{Q}]=[E:\mathbb{Q}(\sqrt[3]2)][\mathbb{Q}(\sqrt[3]2):\mathbb{Q}] = (2)(3) = 6$ E is the field $\mathbb{Q}(\sqrt[3]2,i\sqrt{3})$ I understand why ...
1
vote
1answer
62 views

Proving that “Every non-trivial ring (i.e. with more than one element ) with unity has a maximal ideal” implies axiom of choice is true

I know that assuming axiom of choice or equivalently Zorn's lemma , it can be proved that every non-trivial ring with unity has a maximal ideal (two sided ) . The wiki article on axiom of choice says ...
0
votes
2answers
60 views

Show that $\mathbb{Z}$ and $2\mathbb{Z}$ are not isomorphic as rings.

Show that $\mathbb{Z}$ and $2\mathbb{Z}$ are not isomorphic as rings. My attempt: Suppose $\mathbb{Z}$ and $2\mathbb{Z}$ are isomorphic as rings, Let $\phi: \mathbb{Z} \rightarrow 2\mathbb{Z}$ be the ...
2
votes
1answer
75 views

Find a ring homomorphism $\tau: \mathbb{F} \rightarrow \mathbb{F}$

Just working on some exam prep questions, and I'm a bit stuck on this one. Let $ \mathbb{F} = \{ a + bX + cX^2 | a,b,c \in \mathbb{F}_2 = \{0,1\} \} $ be a ring with the operations: Addition, ...
1
vote
1answer
48 views

Defining a ring homomorphism in proving $|C| \cdot \frac{\chi(C)}{\dim V} $

Lemma irreducible representation $\rho: G \rightarrow GL(V)$, $C$ conjugacy class, then $$|C| \cdot \frac{\chi(C)}{\dim V} $$ is an algebraic integer. In the start of this proof we have: ...
-4
votes
1answer
45 views

Show a subring of the ring of formal power series is finite generated

Let $F$ be a field, and $F[[X]]$ be the ring of formal power series over F. Let $$R=\{a+X^2 f(X):a∈F,f(X)∈F[[X]]\}$$ and let $\alpha\in F[[X]]$. Show the subring of $F[[X]]$ that generated by $R$ and ...
-2
votes
2answers
43 views

Describe all extensions of the identity map of $\mathbb{Q}$ to an isomorphism mapping $\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$

Here is the full question : Describe all extensions of the identity map of $\mathbb{Q}$ to an isomorphism mapping $\mathbb{Q}(\sqrt[3]{2},\sqrt{3})$ onto a subfield of the algebraic closure of ...
0
votes
1answer
33 views

Showing that a divisor of zero in a commutative ring with unity can have no multiplicative inverse" [duplicate]

A divisor of zero in a commutative ring with unity can have no multiplicative inverse. I don't understand why this statement is true. So for $a,b$ in the ring, $ab=0$ by zero divisor. How can ...
3
votes
1answer
23 views

Regard naturally as modules.

Suppose that $I$ is a two-sided ideal in the ring $R$, and that $M$ is a module over the quotient ring $R/I$. Why can we naturally regard $M$ as a $R$-module that is annihilated by $I$? Conversely, ...
0
votes
1answer
21 views

For a cyclotomic polynomial with p = 5, prove that $G(Q(\alpha)/Q)$ is isomorphic to $C_4$ [duplicate]

I'm currently self-studying and would like to understand automorphisms in relation to splitting fields. Could someone please help me with this question? I'm trying explicitly to write down each ...
0
votes
1answer
24 views

Describing the generator of an evaluation map

I'm having a lot of trouble approaching this question: Let $a \in \mathbb{C}$ and $S:\mathbb{Z}[x] \rightarrow \mathbb{C}$ be a map where $S(f(x)) = f(a)$. Is ker($S$) principal? Describe the ...
0
votes
0answers
19 views

Ascending Chain Condition

In the Adams - Loustaunau book about Gröbner bases, I found in the definition of noetherian ring that a ring satisfies the following condition is a noetherian ring: If $I_1\subseteq I_2\subseteq ...
2
votes
1answer
42 views

Other definition for a local ring

Suppose $R$ a ring with 1 and let $U (R)$ denote its invertible elements. If $(M = R \setminus U(R), +)$ is a group, show that $M$ is a left ideal of $R$. I know that $U (R) M \subseteq M$. But ...
0
votes
0answers
34 views

Is Spec R compact? [duplicate]

And if so, why? I'm having some trouble with this. I know that the $D_{f}$ (set of primes not containing $f$) are the open sets and form a basis for the Zariski topology; but I do not know how to go ...
1
vote
1answer
48 views

Show that $R[x]/\langle x \rangle \cong R$ in a ring with 1

Let $R$ be a ring with 1. Show that $R[x]/\langle x \rangle \cong R$. I dont know how i can prove it. Can someone give me a hint on how to approach this problem? Also how would I use the first ...
1
vote
3answers
66 views

Number of elements in a ring such that $x^2=1$

How many elements,$x,$ in the ring $\mathbb Z/95 \mathbb Z$ such that $x^2=1?$ Hints please... Thanks in advance...