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
votes
1answer
210 views

What does it mean for a ring to be unital?

What is the category of unital rings like? I only know that it no more has a terminal object. But what about the products and coproducts? Are they as usual, different or nonexistent? In Gelfand ...
1
vote
3answers
221 views

Let $I$ be an ideal generated by a polynomial in $\mathbb Q[x]$. When is $\mathbb Q[x] / I$ a field?

I was looking at my old exam papers and I was stuck on the following problem: Let $I_1$ be the ideal generated by $x^4+3x^2+2$ and $I_2$ be the ideal generated by $x^3+1$ in $\mathbb Q[x]$. If ...
2
votes
2answers
618 views

What is the kernel of the evaluation homomorphism?

I'm studying Sharp's Steps in Commutative Algebra, and I need a hint how to proceed with this exercise in the page 26: First of all, I didn't understand even the notation, what did the author mean ...
0
votes
1answer
51 views

Example in which the first part of the definition of UFD fails to hold

There is this example in which the first part of the definition of UFD (that is the existence of factorisation) fails to hold that I don't quite understand. Let $R=\mathbb{R}[X_1,X_2,\dots]$, and ...
1
vote
4answers
92 views

Can we see a ring $R$ as a subring of $S^{-1}R$?

I know that we can consider an integral domain $D$ as a subring of its quotient field, I'm wondering why we can't consider any commutative ring with identity as a subring of $S^{-1}R$ identifying ...
1
vote
1answer
41 views

Why is $N(x)=\pm1$ in this problem?

I encountered a proof but there is one step in the proof that I don't really understand. To summarise I just write some portions of the proof: In the ring $\mathbb{Z}[\sqrt{-5}]$, define a function ...
2
votes
2answers
209 views

Two problems about rings.

Somebody can to help me in such exercices: (1) A ring R such that $a^2 = a$ for all $a\in R$ is called a Boolean ring. Prove that every Boolean ring R is commutative and $a + a = 0$ for all $a ...
7
votes
6answers
494 views

Why $\mathbb{R}[X]/(X^2+1)\cong\mathbb{C}$?

There is this isomorphism in my notes but there is no explanation. So I tried to reason myself but still not convincing enough, or my reasoning may even be wrong. I will appreciate if anyone is ...
3
votes
1answer
77 views

How can be possible $\cap R_M=R$?

I'm doing this question in Hungerford's book: I didn't understand how can be possible this intersection be equal to $R$, because $R_M$ is $S^{-1}R$ with $S=R-M$, maybe the author means they are ...
0
votes
1answer
114 views

If $s^n\in S$ and $S$ is a multiplicative subset of a comutative ring, then $s\in S$

I'm trying to solve this question: Let $S$ be a multiplicative subset of a commutative ring $R$ with identity. If $I$ is an ideal in $R$, then $S^{-1}\left(\operatorname{Rad}I\right) = ...
9
votes
2answers
299 views

$K[x_1, x_2,\dots ]$ is a UFD

I wonder about how to conclude that $R=K[x_1, x_2,\dots ]$ is a UFD for $K$ a field. If $f\in R$ then $f$ is a polynomial in only finitely many variables, how do I prove that any factorization ...
1
vote
0answers
36 views

Shouldn't Gallian state $\Leftarrow$ for $a\ne0?$

Gallian text says in an integral domain $D,$ $a$ is a prime element $\iff(a)$ is a prime ideal of $D.$ Showing $\Rightarrow$ is easy. But I can see that $(0)$ is a prime ideal of $D$ (since $D/(0)$ ...
4
votes
2answers
157 views

When a group algebra (semigroup algebra) is an Artinian algebra?

When a group algebra (semigroup algebra) is an Artinian algebra? We know that an Artinian algebra is an algebra that satisfies the descending chain condition on ideals. I think that a group ...
0
votes
2answers
304 views

Let $R$ be a ring with $1$. a nonzero proper ideal $I$ of $R$ is a maximal ideal iff the $R/I$ is a simple ring.

Let $R$ be a ring with $1$. Prove that a nonzero proper ideal $I$ of $R$ is a maximal ideal if and only if the quotient ring $R/I$ is a simple ring. My attempt:- $I$ is maximal $\iff$ $R/I$ is a ...
5
votes
1answer
192 views

Grothendieck group of a symmetric monoidal category is a lambda ring?

I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a ...
9
votes
1answer
324 views

Show that $\left(1+\dfrac{x}{n}\right)^n \to e^x$ as $n \to \infty$ in a normed ring $R$

I had an interesting discussion yesterday with one of my friends (I think he is a member here, am I right?). He claimed that $$\left(1+\dfrac{x}{n}\right)^n \to e^x$$ basically in any normed ring ...
3
votes
1answer
142 views

Question in Hungerford's book

I'm trying to solve this question in Hungerford's Algebra I didn't use the corollary: And I used this map: $g:S^{-1}R_1\to S^{-1}R_2$, $g(r/s)=f(r)/f(s)$. I'm wondering how to prove using the ...
7
votes
0answers
87 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
5
votes
1answer
158 views

Finding the completion of a coordinate ring

Consider $A=\mathbb C[x,y]/(y^2-x(x+1))$, and consider the $\mathfrak m$-adic completion, where $\mathfrak m =(x,y)$. I want to show that this completion is isomorphic to $\mathbb C[[u,v]]/(uv)$, ...
5
votes
1answer
1k views

Ring homomorphisms $\mathbb{R} \to \mathbb{R}$.

I got this question in a homework: Determine all ring homomorphisms from $\mathbb{R} \to \mathbb{R}$, also prove that the only ring automorphism of $\mathbb{R}$ is the identity. I know that ...
5
votes
1answer
289 views

Let A and B be $n \times n$ real matrices with same minimal polynomial.

Let $A$ and $B$ be $n \times n$ real matrices with same minimal polynomial. Then (i) $A$ is similar to $B$. (ii) $A-B$ is singular. (iii) $A$ is diagonalizable if $B$ is so. (iv) $A$ and $B$ ...
3
votes
2answers
95 views

How to understand ideals in $F$, which is a finite commutative ring with $1$

I do not fully understand about ideals in finite rings, and I have to choose the correct answer to the following: If $F$ is a finite commutative ring with $1,$ then (i) Each prime ideal is a maximal ...
2
votes
1answer
77 views

do you need a Noether ring for Noetherian Theorem?

just wondering if a Noetherian ring has any relation to the conservation law of Noether's Theorem? I thought I read the universal enveloping algebra can fall under a Noetherian ring, and was ...
1
vote
1answer
124 views

Proving that $\mathbb{Z}[i]$ is a noetherian ring

Claim: the ring $\mathbb{Z}[i]$ is a noetherian ring My proof 1) $\mathbb{Z}[i]$ is a finitely generated $\mathbb{Z}$-module. 2) $\mathbb{Z}$ is a noetherian ring. 3) Every finitely generated ...
18
votes
1answer
251 views

How 'commutative' can a non-commutative ring be?

Let $R$ be a finite non-commutative ring. Let $P(R)$ be the probability that two elements chosen uniformly at random commute with each other. Consider the value $$S=\sup_RP(R)$$ where the supremum ...
10
votes
2answers
218 views

Directly indecomposable rings

Is every ring the (possibly infinite) direct product of directly indecomposable rings? I believe the answer is no, but I'm not positive and don't know any explicit examples. A reduction: If $R$ is ...
2
votes
2answers
151 views

Every principal ideal domain satisfies ACCP.

Every principal ideal domain $D$ satisfies the ACCP. Proof. Let $(a_1) ⊆ (a_2) ⊆ (a_3) ⊆ · · ·$ be a chain of principal ideals in $D$. It can be easily verified that $I = \displaystyle{∪_{i∈N} ...
3
votes
1answer
100 views

Let $f (x) \in \mathbb{Z}[x]$ be irreducible. Prove that $f(x)$ is primitive.

Let $f (x) \in \mathbb{Z}[x]$ be irreducible. Prove that $f(x)$ is primitive. My thought:- Let $f(x)$ is not a primitive.Since $f(x)$ is not a primitive we can assume $\deg(f(x))>1$.Then there ...
16
votes
2answers
474 views

Example of unital non-commutative ring with $(ab)^2=(ba)^2$ for all $a,b$

I'm trying to exhibit a unital, non-commutative ring $R$ such that $(ab)^2=(ba)^2$ for all $a,b\in R$. This is an exercise out of Herstein's Topics in Algebra. In the previous exercise, I showed ...
3
votes
4answers
127 views

A domain with only a (non-zero) prime ideal

What is an example of a domain $A$ such that Spec$A=\{(0),\mathfrak p\}$? For instance one could find a principal ideal domain that is also a local ring but I can't imagine such a ring.
2
votes
2answers
166 views

Multiplicity of the simple $R$-module $M$ in the semisimple ring $R$

I'm confused about the conclusion of Wedderburn's structure theorem for semisimple rings. Let's consider the special case where $R=M^n$ as modules for some simple module $M$. Wedderburn's theorem says ...
2
votes
0answers
107 views

In $ℤ/Nℤ$, which units are successors to zero divisors?

What are the units $x$ in $ℤ/Nℤ$ of the form $x = 1 + \overline{kd}$ for a divisor $d$ of $N$ and $k ∈ ℤ$, i.e. $$U_N[d] := \{x ∈ (ℤ/Nℤ)^×;\; ∃ k ∈ ℤ : x = 1 + \overline{kd}\} = \ker \big((ℤ/Nℤ)^× → ...
12
votes
3answers
278 views

Clearly, $3 \in \mathbb{Z}$ is not a unit, because $1/3 \notin \mathbb{Z}$. What theorem does this kind of reasoning appeal to?

Intuitively, we may conclude that $3$ is not a unit in $\mathbb{Z}$ simply by observing that $1/3 \notin \mathbb{Z}$. However, what does this reasoning actually appeal to??? Is it true that: ...
2
votes
1answer
62 views

If every irreducible element in $D$ is prime, then $D$ has the unique factorization property.

Suppose every irreducible element in a domain $D$ is prime. I'm trying to prove this implication: In a integral domain $D$, if $a=c_1c_2...c_n$ and $a=d_1d_2...d_m$ ($c_i,d_i$ irreducible), ...
4
votes
0answers
247 views

Ideals in Gaussian integers

Let $R:=\mathbb{Z}[i]$. Prove that every nonzero prime ideal $\mathfrak{P}$ of $R$ belongs to one of the following families: 1) $\mathfrak{P}=(1+i)R$ 2) $\mathfrak{P}=(a+bi)R$ where ...
2
votes
5answers
3k views

(Simple) Examples on Non Commutative Rings

Looks like it is easier to find example of commutative rings rather than non commutative rings. Prabably the easiest examples of the former are $\mathbb{Z}$ and $\mathbb{Z}_n$. We can find ...
3
votes
1answer
212 views

Classification of prime ideals in $\mathbb{Z}[i]$

It's a couple of days that i'm struggling with this answer, which i'd like very much to understand. I recall briefely what is the problem: I want to classify all prime ideals of $\mathbb{Z}[i]$. The ...
2
votes
1answer
133 views

$p$ is irreducible if and only if the only divisors of $p$ are the associates of $p$ and the unit elements of $R$

Let $R$ be an integral domain and $p ∈ R$ be such that $p$ is nonzero and a nonunit. Then $p$ is irreducible if and only if the only divisors of $p$ are the associates of $p$ and the unit elements of ...
0
votes
1answer
73 views

Why Integral domains haven't an unified definition? [duplicate]

We can define integral domains as: rings without zero divisors commutative rings without zero divisors commutative rings with identity and without zero divisors I don't know why integral domains ...
5
votes
1answer
341 views

Prime ideals in quotients of polynomial ring over finite field

Reading a book, i found this argument ($\mathbb{F}_2$ is the field with 2 elements): consider the quotient ring: $$\mathbb{F}_2[x]/(x+1)^2$$ Then it has only one prime ideal, namely the following: ...
1
vote
3answers
2k views

In a integral domain every prime element is irreducible

I'm trying to understand a proof of Hungerford's book which says that in a integral domain every prime element is irreducible: I didn't understand why this implication $p=ab\implies p|a$ or $p|b$, ...
0
votes
1answer
67 views

Describe ideals of a ring

I have difficulties with a task in linear algebra. R is a ring and R = {a, b, c, d} These are the tables for + and . in R: ...
2
votes
2answers
181 views

Local fields and infinite extensions, basic questions

Notation throughout: Let $K$ be a discrete valuation field and $L/K$ an infinite (not necessarily Galois) extension of $K$. 1) How can/does one define a ramification index $e(L/K)$ for $L/K$? It ...
1
vote
2answers
74 views

$\operatorname{\mathcal{Jac}}\left( \mathbb{Q}[x] / (x^8-1) \right)$

$\DeclareMathOperator{\Jac}{\mathcal{Jac}}$ Using the fact that $R := \mathbb{Q}[x]/(x^8-1)$ is a Jacobson ring and thus its Jacobson radical is equal to its Nilradical, I already computed that $\Jac ...
0
votes
2answers
61 views

How to prove this function is surjective

I'm trying to solve this question: In order to solve this question above, I found this function: $r/w\mapsto (r/s)/(w/s)$ such that $w/s\in T$, I almost proved this map is an isomorphism, I'm stuck ...
4
votes
2answers
333 views

Is the completion of a commutative Noetherian local ring Noetherian?

Maybe for some straightforward, but not for me: Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring with maximal ideal $\mathfrak{m}$. Why is the completion $\widehat{R}$ of $R$ with ...
3
votes
1answer
145 views

Localization and a particular exact sequence

Look at this proposition: Let $R$ be a commutative ring with unity, and let $f_1,\ldots,f_n\in R$ generate the unit ideal in $R$. Then the following sequence is exact: $$0\longrightarrow ...
2
votes
3answers
59 views

Prove that certain elements are not in some ideal

I have the following question: Is there a simple way to prove that $x+1 \notin \langle2, x^2+1\rangle_{\mathbb{Z}[x]}$ and $x-1 \notin \langle2, x^2+1\rangle_{\mathbb{Z}[x]}$ without using the ...
3
votes
1answer
236 views

Is the quadratic character, unique multiplicative character over $\mathbb Z_{p^n}$, for odd $p$?

Let $p$ be odd and $\mathbb Z_{p^n}$ denote the ring of integers modulo $p^n$. Let the quadratic character, $\eta$, be the function defined on $\mathbb Z_{p^n}^*$ (multiplicative group of $\mathbb ...
3
votes
2answers
128 views

Basic ring isomorphism question

Let $R=\mathbb{Z}[x], I=(x^2+1,x+1).$ Prove that $R/I \cong \mathbb{Z}[i]/(i+1) \cong \mathbb{Z}/2\mathbb{Z}$. I am confused with the rather messy looking of $R/I$. My first step is to define the ...