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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2
votes
1answer
25 views

If $a \ne b$ in a ring $R$ satisfy $a^3 = b^3$ and $a^2b = b^2a$, show that $a^2 + b^2$ is not a unit.

If $a \ne b$ in a ring $R$ satisfy $a^3 = b^3$ and $a^2b = b^2a$, show that $a^2 + b^2$ is not a unit. So I am thinking that I should be able to do this by contradiction. So if I assume there is some ...
1
vote
1answer
45 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...
0
votes
2answers
18 views

If char$R=n$, show that $\mathbb{Z}1_R\cong \mathbb{Z}_n$.

Let $1_R$ be the identity of a ring $R$ and let $\mathbb{Z}1_R=\{k1_R\mid k\in\mathbb{Z}\}$. If char$R=n$, show that $\mathbb{Z}1_R\cong \mathbb{Z}_n$. So my thought is I just have to think of some ...
2
votes
1answer
28 views

Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
1
vote
3answers
27 views

If $S$ and $T$ are subrings of $R$, is $S+T$ a subring of $R$?

If $S$ and $T$ are subrings of $R$, is $S+T=\{s+t\mid s\in S, t\in T\}$ a subring of $R$? So I think that $S+T$ is a subring, but I am getting stuck trying to prove it. Clearly since $S$ and ...
1
vote
2answers
20 views

Clarify what “inclusion preserving” means in lattice isomorphism theorem

I'm working through Dummit and Foote right now. The lattice isomorphism theorem is stated as follows: "Let I be an ideal of a ring R. The correspondence $A \leftrightarrow A/I $ is an inclusion ...
0
votes
0answers
23 views

why is the collection of all finite subsets of $\mathbb{R}$ not a $\sigma-ring$

It says the definition of a $\sigma-ring$ is if $A,B \in \mathcal R$ then $A \setminus B \in \mathcal R$ and if $ A_{n} \in \mathcal R \forall n \in \mathbb{N}$ then $\cup_{1}^{\infty}A_{n} \in ...
3
votes
1answer
15 views

Relationship between operations of a ring

Is there any requirement that the two operations of a ring have to be related to each other, excluding the requirement of distributivity? We all know from grade school that multiplication of integers ...
0
votes
1answer
10 views

factors in polynomial rings with field coefficients

I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following: Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if ...
1
vote
3answers
27 views

A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime

I'm trying to prove that when $p ≡ 1 \pmod3$ is a prime, $p$ is reducible over the Eisenstein integers, and I've gotten to the point where, provided $p\,|\,u^2 - u + 1$ for some integer $u$, then $p$ ...
0
votes
1answer
32 views

Quotient of maximal and prime ideals

Given that $I, J$ are ideals in $R$, $I$ is maximal or prime, do we have that $I/J$ is maximal in $R/J$? $I/J$ is prime in $R/J$? I think it is true but don't see how it works.
2
votes
3answers
69 views

Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$? [duplicate]

I understand geometrically why the Gaussian integers modulo $i+1$ is $\mathbb{Z}_{2}$, using lattices. Is there an algebraic isomorphism construction, however?
0
votes
2answers
47 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} ...
0
votes
0answers
21 views

The coordinate ring of $\varepsilon: xy-1=0$ [duplicate]

I want to show that the coordinate ring $\mathbb{R}[x,y]/\mathbb{R}[\varepsilon]$ of $\varepsilon: xy-1=0$ is not isomorphic with the polynomial ring of one variable $\mathbb{R}[x]$. To me this is ...
0
votes
1answer
11 views

Addition of fractions in z11

compute 3/5+2/7+1/6 in Z11. Please give me a hint on how to go about it. I have created a table for Z11 but unsure of the next step.
0
votes
1answer
81 views

Algebraic element - integral domain

Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k \in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that $\overline{\overline{L}}=\overline{L}$. ...
3
votes
2answers
50 views

Prove or disprove : the number $3+2\sqrt{-2}$ is irreducible in the ring $\mathbb{Z}[\sqrt{-2}]$

Prove or disprove: the number $3+2\sqrt{-2}$ is irreducible in the ring $\mathbb{Z}[\sqrt{-2}]$. I think it is sufficient to show that each element (except $0$) in $\mathbb{Z}\sqrt{D}$ with $D ...
0
votes
1answer
12 views

Principal Ideal Domain $R$ and ideal $J\neq 0$ so that that $R/J$ have a finite number of ideals. [duplicate]

Let $R$ a un Principal Ideal Domain(PID) and $J\neq 0$ a ideal of $R$. Show that $R/J$ have a finite number of ideals.
1
vote
1answer
22 views

An isomorphism between product of number fields, contains the same number of factors

Suppose that we have an isomorphism of rings $$f:K_1\oplus\cdots\oplus K_r\to K'_1\oplus\cdots\oplus K'_s,$$ with $K_i$'s and $K_j'$'s are a number fields, the sum and the product are componentwise. ...
0
votes
1answer
42 views

Which of the following are true about the ring of continuous real valued functions C[0,1]

Let $C[0,1]$ be the space of continuous real-valued functions on the interval $[0,1]$. This is a ring under point-wise addition and multiplication. Which of the following are true: (a) For any $x ∈ ...
0
votes
1answer
43 views

non-zero divisors in a ring

I am asked to show the following: $ab$ is a non-zero divisor of $R$ if and only if $a$ and $b$ are both non-zero divisors of $R$. $\Rightarrow)$ Suppose $ab$ is a non-zero divisor of R. Then ...
1
vote
0answers
29 views

When is the ideal generated by 2 elements equal to the sum of the 2 ideals

Is it true in general that (a,b)=(a)+(b)? I would suppose that (a)+(b)$\subset$(a,b) and i believe the reverse containment should hold as well, i just can't seem to fit the pieces together.
6
votes
1answer
59 views

Inverses of elements in group algebras

If $G$ is a finite group whose elements are $g_1,\ldots,g_n$ and let $F$ be the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. We define a vector space over $F$ with ...
2
votes
1answer
41 views

Proving that for an integral domain $R$, $y\in (x)\iff (y)\subseteq (x)$.

I am trying to prove the following statement. Let $R$ be a integral domain. Then for all $x,y\in R$ we have $$x\mid y\iff y\in(x)\iff (y)\subseteq (x).$$ Note that $(x)$ denotes the principal ...
-2
votes
2answers
82 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 ...
3
votes
1answer
143 views

Group ring C[Z/n] and Artin-Wedderburn decomposition

I am trying to answer the following questions, which I assume follow on from eachother each other; Write $\mathbb C$[$\mathbb Z$/n] as a product of simple rings. For abelian groups $G_1$, $G_2$, ...
2
votes
1answer
98 views

To find the nilpotent elements of $\Bbb Z_n$ and also the number of nilpotent elements of $\Bbb Z_n$.

I am trying to find the nilpotent elements of $\Bbb Z_n$ and also the number of nilpotent elements of $\Bbb Z_n$. My Try: If $\bar a$ be a nilpotent element then there exists a $k \in \Bbb Z$ such ...
3
votes
1answer
55 views

Derived categories of filtered modules

For a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which ...
1
vote
1answer
40 views

If there exists a vertex of $ \Gamma_{2}(R)\setminus J(R) $ which is adjacent to every other vertex then $ R \cong \mathbb{Z}_{2}\times F$

I am reading the research paper Comaximal Graph of Commutative Rings by H.R. Maimani, M. Salimki, A. Sattari, S. Yassemi. In this paper, $ R $ denotes a commutative ring with the identity element. $ ...
0
votes
2answers
37 views

About radical of $(I,x)$ with $x$ irreducible

Let $I$ be a proper ideal of a polynomial ring $A$ and $x \in A$ an irreducible element. In a theorem of commutative algebra I will use the fact that, in this hypothesis, holds the following ...
1
vote
2answers
313 views

Ideals of formal power series ring

I need help understanding the following solution for the given problem. The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i ...
0
votes
1answer
88 views

$\mathbb R[X]/\langle X^4-1\rangle \cong \mathbb R \times \mathbb R \times \mathbb C$

I am trying to prove the isomorphism $\mathbb R[X]/\langle X^4-1\rangle \cong \mathbb R \times \mathbb R \times \mathbb C$. I will write what I did so you can help me from there. First notice that ...
1
vote
1answer
89 views

Basis of the ring $B=\operatorname{End}_R\left(R^{(\mathbb N)}\right)$

Let $B=\operatorname{End}_R\left(R^{(\mathbb N)}\right)$. Define $u,v \in B$ as $$u(e_{2_{i+1}})=0,\, u(e_{2_i})=e_i\\v(e_{2_{i+1}})=e_i,\,v(e_{2_i})=0$$ Prove that $\{u,v\}$ is a basis of $B$ ...
2
votes
1answer
464 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...
1
vote
1answer
194 views

Noetherian group rings [on hold]

I'm asking for an example of a finitely generated amenable group $G$ and a field $K$, such that the group ring $K[G]$ is not Noetherian. Is it also possible to find a finitely generated amenable ...
5
votes
1answer
899 views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
0
votes
0answers
44 views

Brauer Group - A measure of complexity?

I have seen many authors state that the Brauer Group in some way measures the complexity of a field. I've convinced myself that the Brauer group of the reals is Z/2Z, and that the Brauer group of an ...
4
votes
2answers
94 views

Existence of nontrivial unit in $\mathbb{Q}[G]$, where $G$ is finite.

Suppose $G$ is a finite group of order $|G|>1$, and $\mathbb{Q}[G]$ is the group ring. I'm curious about an example of a nontrivial invertible element, i.e., one that is not of the form $ag$, ...
9
votes
2answers
413 views

Can the product of two non invertible elements in a ring be invertible?

Let $A$ be a unitary ring. The question is simply: can the product of two non invertible elements in $A$ be invertible? I proved that the answer is negative if $A$ does not have zero divisors, ...
1
vote
1answer
62 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
6
votes
2answers
2k views

Describe all ring homomorphisms

Describe all ring homomorphisms of: a) $\mathbb{Z}$ into $\mathbb{Z}$ b) $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$ c) $\mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z}$ d) How many ...
2
votes
1answer
1k views

Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$ [duplicate]

Suppose $R$ is a commutative ring. Show that every ideal of $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$. I have spent 30 minutes on this question and I still got nowhere. Can ...
268
votes
0answers
12k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
4
votes
3answers
356 views

can the square of a proper ideal be equal to the ideal

Let $R$ be a ring, commutative with $1$, let $\mathfrak{i}$ be an ideal, not the whole ring. In general $\mathfrak{i}^2\subseteq\mathfrak{i}$. Can this inclusion be an equality, or it is always a ...
10
votes
2answers
845 views

Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
11
votes
3answers
2k views

What are applications of rings & groups?

I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be ...
0
votes
2answers
252 views

Why is $\mathbb F_2 [X] / \langle X^2 + X + 1 \rangle \cong \mathbb F_4$?

What's the easiest way to see why $\mathbb F_2 [X] / \langle X^2 + X + 1 \rangle \cong \mathbb F_4$? The polynomial is irreducible in $\mathbb F_2 [X]$, but that's about the only observation I've ...
7
votes
2answers
3k views

When is a product of two ideals strictly included in their intersection?

Let $I,J$ two ideals in a ring $R$. The product of ideals $IJ$ is included in $I \cap J$. For example we have equality in $\mathbb{Z}$ if generators have no common nontrival factors, in a ring $R$ ...
5
votes
0answers
156 views

An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...
25
votes
3answers
614 views

A Laskerian non-Noetherian ring

A Laskerian ring is a ring in which every ideal has a primary decomposition. The Lasker-Noether theorem states that every commutative Noetherian ring is a Laskerian ring (as an easy consequence of the ...