This tag is for questions about rings, which are a type of structure studied in abstract algebra and algebraic number theory.
-3
votes
0answers
88 views
Idempotent isomorphic to $1$
Let $R$ be a ring with unit element $1$. An idempotent element $e \in R$ is called isomorphic to $1$ if there are idempotent elements $u, v \in R$ such that $vu=1$ and $uv=e$ and this is written $e ...
1
vote
2answers
25 views
Help to understand the ring of polynomials terminology in $n$ indeterminates
In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner:
After the author defines the operations in this ring with a theorem:
...
3
votes
1answer
85 views
If $a^3=a$ in a ring, prove: the ring is commutative [duplicate]
Let $R$ be a ring, not necessarily with a unit element. $R$ is not necessarily integral.
If for any $a \in R$, $a^3=a$, prove: $R$ is commutative:
Any $a, b \in R$, $ab=ba$.
My efforts on it:
I can ...
4
votes
4answers
82 views
The elements of $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$
I'm really confused with this one...
How can I determine the elements of the module $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$? Or its cardinality?
Does ...
-6
votes
2answers
58 views
Help with Theorem III.3.11 in Hungerford's algebra book
I need help to prove part (i) of this theorem which I couldn't prove.
Any help would be appreciated. Thanks in advance.
4
votes
3answers
47 views
Factorize in R[x]
I have the polynomial $x^8+1$, I know that there's no root for solve this in $\Bbb R[x]$ but i want to factorize this to the minimal expression. This is possible or this is irreducible?
2
votes
2answers
63 views
Why the terms “unit” and “irreducible”?
I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition
Maybe historical reasons?
For example, I suppose the second ...
3
votes
2answers
55 views
Exercise about prime ideals
Let $A$ be a ring. Prove that the following conditions are equivalent:
$i)$ All ideals $I \subsetneq A$ are prime.
$ii)$ The set of all ideals of $A$ is totally ordered by
inclusion and all ideals ...
4
votes
1answer
59 views
The Baer-McCoy (a.k.a. prime) radical of $A$
Let $B$ a ring and let $A$ a subring of $B$.
Show that $A\cap \mathrm{Nil}_{*}(B)\subset \mathrm{Nil}_{*}(A)$.
If $A$ is contained in the center of $B$, show that $A\cap ...
0
votes
1answer
17 views
Questions on (subring/ submodule) of a graded (ring/ module)
I have a question which seems a bit silly...
If we have $R$ a graded ring, does it follow that every subring of $R$ is also graded?
Because I have a problem here as such: I have a graded ring $R$ ...
2
votes
0answers
31 views
Find the factorization of the polynomial as a product of irreducible [duplicate]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$
Testing with the simplest possible root in this case, $P(1)=0$
Applying the ...
1
vote
1answer
36 views
Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$
Testing with the simplest possible root in this case, $P(1) = 0$
...
6
votes
1answer
49 views
Calculating in quotient ring of $\mathbb{R}[X]$
Part of an old Oxford exam (1992 A1)
We want to find which elements of the quotient ring $\mathbb{R}[X]/(x^3-x^2+x-1)$ are equal to their own square.
Now, we note first that ...
5
votes
1answer
32 views
Generalising the Chinese Remainder Theorem
We have that for $I,J$ ideals of some ring $R$ with $R=I+J$, $$\frac{R}{I\cap J} \cong \frac{R}{I} \times \frac{R}{J}$$
My question is whether the analogous expression for three ideals $I,J,K$ where ...
5
votes
1answer
36 views
Express $4+\sqrt{-2}$ as a product of irreducibles
This is part of an old Oxford Part A exam paper. (1992 A1)
Suppose we equip $R=\mathbb{Z}[\sqrt{-2}]$ with the Euclidean function $d$ defined by $$d(m+n\sqrt{-2})=|m+n\sqrt{-2}|^2$$
I want to ...
1
vote
4answers
56 views
Subrings of $\mathbb{Q}$
Let $p$ be prime. Suppose $R$ is the set of all rational numbers of the form $\frac{m}{n}$ where $m,n$ are integers and $p$ does not divide $n$.
Clearly then $R$ is a subring of $\mathbb{Q}$.
I now ...
4
votes
3answers
60 views
example of a flat but not faithfully flat ring extension
I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
5
votes
3answers
80 views
Quotient rings of Gaussian integers $\mathbb{Z}[i]/(2)$, $\mathbb{Z}[i]/(3)$, $\mathbb{Z}[i]/(5)$,
I am studying for an algebra qualifying exam and came across the following problem.
Let $R$ be the ring of Gaussian Integers. Of the three quotient rings
$$R/(2),\quad R/(3),\quad R/(5),$$
one ...
4
votes
2answers
94 views
Irreducibility of $x^n-x-1$ over $\mathbb Q$
I want to prove that
$p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible.
My attempt.
GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
4
votes
1answer
57 views
How to compute Nakayama functor explicitly?
I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
0
votes
1answer
43 views
Do there exist any polynomial rings with nonzero prime ideals that are not maximal?
I know that with $F$ a field, $F[x]/(f(x))$ is a field iff $f(x)$ is irreducible in $F$. Due to the fact that in a UFD irreducible elements are necessarily prime, we would have that $(f(x))$ is both ...
0
votes
2answers
52 views
Rings | Homomorphisms | Units
Question
Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is
a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$.
Attempt
...
2
votes
2answers
86 views
Under what conditions is a zero divisor element $a$ in commutative ring $R$ nilpotent?
Suppose that $R$ is a commutative ring with identity $1$
Let $a\in R$ with $ab=0$ for some $b\ne0$.
Under what conditions $a$ must be also nilpotent?
2
votes
2answers
22 views
length of sum of two submodule
Let $M$ be a $R$-module with finite length and $K$ and $N$ be a submodule of $M$. Prove that
$l(K+N)+l(K\cap N)=l(K)+l(N)$.
My proof: First, by assuming that $K\cap N=\{0\}$, we can conclude that ...
6
votes
1answer
44 views
Given $G$, when can we find a division ring $R$ with $R^*=G$?
This is motivated by a characterization of finite cyclic groups, in which one proves
Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic.
The proof is ...
1
vote
1answer
43 views
A question in Ring theory [duplicate]
R be a commutative ring with unity and it has exactly one maximal ideal. Then prove that the equation $x^2 =x$ has exactly two solutions.
Show me the right way to solbe this one.
5
votes
0answers
44 views
+100
Amenable group rings embeddable in skew fields
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:
(1) the group ring $K[G]$ is a domain;
(2) $K[G]$ is ...
3
votes
3answers
82 views
Do Boolean rings always have a unit element?
Let $(B, +, \cdot)$ be a non-trivial ring with the property that every $x \in B$ satisfies $x \cdot x = x$.
How does one prove that such a ring $(B, +, \cdot)$ must have a unit element $1_B$? (Or, ...
8
votes
2answers
59 views
If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions
If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions.
I know that $0$ and $1$ are the solutions, but I can't proceed ...
4
votes
2answers
51 views
Is the kernel of a homomorphism from a Boolean ring to $\mathbb{Z}_2$ always a maximal ideal?
Let $(B, +, \cdot)$ be a ring (not necessarily unital!) with the property that every $x \in B$ satisfies $x \cdot x = x$.
How does one show that the kernel of any non-zero homomorphism of rings ...
2
votes
0answers
17 views
Number of left ideals in a simple ring
I'm puzzling over a few algebra questions:
1) Give an example of a simple ring with exactly $12$ non-zero proper left ideals.
For this one I have no idea, I am not good with coming up with examples ...
-5
votes
0answers
46 views
Theory of rings $1$ [closed]
$D$ is a division ring. All the prime ideals and semiprime the $A=T_3(D)$, the ring of upper triangular 3x3 matrix with coefficients in $A$. What would be the ideal?
4
votes
0answers
54 views
Artinian rings are perfect
Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
-5
votes
1answer
56 views
Find all ideals of ring $\mathbb{Z}/m\mathbb{Z}$. [closed]
Find all ideals of ring $\mathbb{Z}/m\mathbb{Z}$.
7
votes
2answers
53 views
Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?
Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
20
votes
4answers
490 views
Why do we study prime ideals?
I hope this isn't an inappropriate question here!
I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
3
votes
2answers
63 views
A noetherian ring whose ideals are idempotent is artinian
I have to prove the folowing:
If $R$ is a Noetherian ring, and for every ideal $I$ of $R$ we have $I = I^{2}$, then $R$ is Artinian.
My first thought was to try to prove that the nilradical of ...
3
votes
1answer
63 views
Is the endomorphism ring of $\mathbb{R}$ self-opposite?
Is the endomorphism ring $End\mathbb{R}$ of the Abelian group $\mathbb{R}$ isomorphic to its opposite ring?
All subrings of a self-opposite ring are self-opposite. By choosing an isomorphism of ...
5
votes
0answers
86 views
An example of a commutative ring in which every primary ideal is prime
It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
2
votes
1answer
54 views
Proving part of the Wedderburn Structure Theorem
I'm having trouble with an exercise from "Noncommutative Algebra," by Farb & Dennis proving part of the Wedderburn structure theorem for semisimple rings.
If $R$ is a semisimple ring and ...
5
votes
1answer
50 views
What can be said about $p\in Spec(R)$ when $R_p$ is a field?
What can be said about $p\in Spec(R)$ when $R_p$ is a field? Especially when $R$ is local noetherian
3
votes
2answers
51 views
$\mathbb{C}[G]$-module homomorphism on finite dimensional modules and finite groups
Nice to meet you folks! I'm currently a grad student reviewing some representation theory of finite groups for prelims next year, and I'm stuck proving a simple statement. Translating the question ...
1
vote
1answer
51 views
Prove that if R and S are nonzero rings then $R\times S$ is never a field.
This question has been asked on here before but I'm looking for some additional insights. The question comes from the section of the Dummit and Foote textbook on the Chinese Remainder Theorem. All of ...
3
votes
0answers
52 views
Integral homomorphism induces a closed map on spectra
I'm trying to prove the following:
Let $f:A\rightarrow B$ is a integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
2
votes
1answer
22 views
Characteristic of commutative semisimple rings?
In one of my questions (Structure of the group ring of a direct product?), a statement is made for a commutative semisimple ring of characteristic $p^t, t\geq1$. Now I don't understand why there ...
3
votes
1answer
33 views
$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives
So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
3
votes
1answer
36 views
Find conditions for there to exist a morphism of rings from $\mathbb{Z}_m$ to $\mathbb{Z}_n$
I know that a necessary and sufficient condition for a ring morphism $\mathbb Z_m\to\mathbb Z_n$ to exist is that $n$ must divide $m$. However, I am having trouble understanding a proof that this ...
7
votes
1answer
95 views
Ring without $1$ where $\forall r\in R$, $\exists$ $n_r > 1$ such that $r^{n_r} = r$, and not all primes are maximal
On my algebra final exam, there was a problem that essentially asked the following:
Let $R$ be a commutative ring such that for all $r\in R$, there exists $n_r\in\Bbb{Z}^{>1}$ with $r^{n_r} = ...
2
votes
3answers
60 views
Spectrum of polynomial ring
In M. Reid's Undergraduate Commutative Algebra, the author states that if $k$ is an algebraically closed field then $\operatorname{Spec}{k[x]} = \{0\} \cup k$ (page 21). Is this correct? Instead, ...
0
votes
1answer
62 views
Prime ideals in commutative ring
Let $R$ be a commutative ring with $1$ (we take $R$ not to be a field for this post). Must $R$ contain at least one prime ideal that is not maximal?
The question is equivalent to the following: For a ...
