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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8
votes
3answers
869 views

How to show that a finite commutative ring without zero divisors is a field?

$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field? I'm just starting with abstract algebra and I'd really appreciate if someone ...
5
votes
3answers
761 views

Maximal ideals in polynomial rings

Let $K$ be a field. Let $\mathfrak{m}$ be an ideal of the polynomial ring $K[x_1,\ldots,x_n]$ and suppose the quotient $\frac{K[x_1,\ldots,x_n]}{\mathfrak{m}}$ to be isomorphic to $K$ itself. I want ...
3
votes
4answers
385 views

Understanding the quotient ring $\mathbb{R}[x]/(x^3)$.

I am having difficulty in understanding exactly the elements of the set $\mathbb{R}[x]/(x^3)$. I'll explain my thought process. The Quotient Ring is the set of additive cosets, so we have that ...
1
vote
1answer
154 views

Properties inherited from $R$ by Laurent polynomials $R[x;x^{-1}]$

I wonder if there is a paper about the conditions going up to Laurent Polynomial rings For example the Laurent polynomial preserves the condition of reversibility of ring R For a ring $R$ ...
4
votes
3answers
479 views

Ring of all continuous functions from reals into reals is not integral domain

Let $R$ be the ring of all continuous functions from the real numbers into the real numbers. Prove that $R$ is not an integral domain. I need help with this. I do not understand this at all and my ...
6
votes
3answers
337 views

Does Euclid lemma hold for GCD domains?

Exercise $ 10 $ of Section $ 3 $ of Chapter III of Hungerford’s Algebra states that if $ R $ is a UFD and if $ a,b $ are relatively prime, then $ a | bc $ implies $ a | c $, something that is easy to ...
1
vote
1answer
48 views

Conjugacy class in matrix ring

Let $M_{2}(\mathbb{R})$ be the ring of $2\times 2$ matrices over the reals and $M_{2}(\mathbb{R})^*$ the set of invertible such matrices. Consider any $A \in M_{2}(\mathbb{R})$ such that $ A^{2}=-I$, ...
1
vote
1answer
57 views

$P$ is prime if and only if for every pair of ideals $I,J$ containing $P$ properly, we have $IJ\nsubseteq P$

I'm trying to show that if $P$ is an ideal of a ring $A$; $A$ not necessarly conmutative, then $P$ is prime if and only if for every pair of ideals $I,J$ containing $P$ properly, we have ...
7
votes
1answer
226 views

Computing the ring of integers of a number field

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
2
votes
2answers
68 views

Why is $S/R$ a ring extension?

If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. ...
1
vote
1answer
47 views

Proof of a linear transformation property

Suppose $\phi:X \rightarrow Y$ is a map of sets and $F$ is a field. Let $\phi^* : F(Y) \rightarrow F(X)$ be a map sending a function $f \in F(Y)$ to a function $\phi^*(f) \in F(X)$ given by ...
9
votes
1answer
223 views

If $I$ is a finitely generated ideal of $A[X]$, is $I\cap A$ necessarily finitely generated for a commutative unital ring $A$?

Let $A$ be a commutative ring with $1$ and $A[X]$ the ring of polynomials in one variable over $A$. Assume $I$ is a finitely generated ideal of $A[X]$. My question is Is $I\cap A$ necessarily ...
0
votes
2answers
57 views

Bijection between $\operatorname{Hom}_{Grps}(G, U(R))$ and $\operatorname{Hom}_{Rings}(\Bbb{Z}G, R)$

I would like to find a bijection between $\operatorname{Hom}_{\mathbf{Grps}}(G, U(R))$ and $\operatorname{Hom}_{\mathbf{Rings}}(\Bbb{Z}G, R)$, where $U(R)$ is the group of units and $\Bbb{Z}G$ is the ...
0
votes
1answer
31 views

Intesection of Ideals

Let P denote prime ideals,an $A_1,A_2$ ideals, i need to know how can i describe $\{P | A_1\subset P\}\cup\{P | A_2\subset P\}$ I mean it is equal to $\{P | A_1\cap A_2\subset P\}$ I think not, ...
3
votes
1answer
174 views

Showing that a ring homomorphism is injective

Let $f\in\mathbb{Z}[X]$ be a monic irreducible polynomial, $\alpha$ a root of $f$ and $k\in \mathbb{Z}$. Show that the map $$\varphi : \mathbb{Z}/f(k)\mathbb{Z} \to \mathbb{Z}[\alpha]/(k - ...
10
votes
7answers
1k views

Applications of the Isomorphism theorems

In my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but I ...
2
votes
2answers
98 views

generators of an ideal, dimension of a vector space

Let $R$ be a local Noetherian ring (maximal ideal $m$, residue field $k$). Suppose $\{x_{1}, \ldots, x_{n}\}$ generate $m$. Is it true that dim$_{k}(m/m^2) \leq n$?
-2
votes
1answer
200 views

$S, T$ be multiplicatively closed sets in the ring $R$, such that $S \subseteq T$ Show that the following are equivalent

Let $S, T$ be multiplicatively closed sets in the ring $R$, such that $S\subseteq T$. Let $\varphi : S^{−1}R \to T^{−1}R$ be the homomorphism which maps each $r/s \in S^{−1}R$ to $r/s$ viewed as an ...
2
votes
1answer
260 views

When is the localization of a module trivial?

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. Let $S^{-1}M$ be localization of $M$, where $S$ is a multiplicatively closed subset of $R$. How to show that $S^{−1}M =0$ if ...
4
votes
1answer
48 views

Kleene algebra without right distributivity?

I'm facing a mathematical structure that has everything of a Kleene algebra (S, +, ., 0, 1, *), except that the multiplication '.' is not right-distributive over the addition '+'. ...
1
vote
2answers
108 views

Subfields of Rings

I am currently working through an undergraduate class in Galois Theory. I have come across a question that I am unsure about. Can a ring that is not a field, have a subring that satisfies the ...
2
votes
3answers
92 views

Show that the ideal of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$is a prime ideal.

Show that the ideal $I$ of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$is a prime ideal, where $k$ is a field. I tried to prove it by contradiction. Suppose $f$ and $g$ are not of ...
1
vote
2answers
143 views

Prove that T is a subring of R

Let $R$ and $S$ be rings and $h$ and $g$ be homomorphisms from $R$ into $S$. Let $T=\{r| r\in R$ and $h(r)=g(r)\}$ Prove that $T$ is a subring of $R$. I understand what the question is asking but ...
4
votes
1answer
662 views

Specific proof that any finitely generated $R$-module over a Noetherian ring is Noetherian.

I have seen a handful of proofs that any finitely generated module over a Noetherian ring is again Noetherian. I'm specifically trying to understand the following proof idea. It goes as this: Observe ...
6
votes
3answers
182 views

Finding the ideals in a ring of fractions

I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, $b$ is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find ...
13
votes
2answers
479 views

Structure of Finite Commutative Rings

Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?
6
votes
3answers
528 views

Ring of order $p^2$ is commutative.

I would like to show that ring of order $p^2$ is commutative. Taking $G=(R, +)$ as group, we have two possible isomorphism classes $\mathbb Z /p^2\mathbb Z$ and $\mathbb Z/ p\mathbb Z \times \mathbb ...
2
votes
2answers
190 views

Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$?

Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$. I know that $\mathrm{char}(R) =2$ $R$ has $2^k$ elements for some ...
1
vote
1answer
2k views

The (Jacobson) radical of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. Let $\Omega$ be the set of all maximal ideals of $R$. Prove that $\operatorname{Rad}(M)=\bigcap_{\mathfrak m\in \Omega}\mathfrak mM$, where ...
2
votes
2answers
126 views

How can I write $6$ as products of irreducibles in the Gaussian Integers $\mathbb{Z}[i]$?

Moreover, how can I prove that $2+$i and $1+i$ are irreducibles?
1
vote
1answer
283 views

Irreducible elements of $\mathbb{Z}[i\sqrt{5}]$

Is there any neat way to show that $9$ and $3-3i\sqrt{5}$ are irreducible elements of $\mathbb{Z}[i\sqrt{5}]$, while $1+4i\sqrt{5}$ and $5-2i\sqrt{5}$ are reducible?
4
votes
1answer
301 views

When is the maximal ideal of a zero-dimensional local non-noetherian commutative ring nilpotent?

Let $R$ be a non-Noetherian local commutative ring with identity such that it is of Krull-dimension zero. I am wondering if there are conditions which will force the maximal ideal to be nilpotent.
3
votes
1answer
133 views

Existence of a ring automorphism

Let $A$ be a commutative ring, $B$ a commutative ring that is also an $A$-algebra of finite presentation. Let $f_1$ and $f_2$ be two elements of $B$ such that $(f_1) + (f_2) = B$ and as $A$-modules, ...
2
votes
3answers
71 views

Existence of a certain R-module homomorphism

Let $R$ be a ring with identity and $A,B,C$ be $R-$modules. Let $f:A\rightarrow C, g:B\rightarrow C$ be $R$-module homomorphisms such that $g$ is surjective. Does there exist an $R-$module ...
0
votes
2answers
109 views

An example of a divisible ideal

Let $R$ be a commutative ring (other than a field) with identity. I am looking for an example of a divisible ideal of $R$.
3
votes
1answer
91 views

Division ring which is algebraic over $\mathbb R$

I am looking for the proof of the following Theorem. If $L$ is a division ring algebraic over $\mathbb R$, then $L$ is $\mathbb R$-isomorphic to $\mathbb R$ or $\mathbb C$ or $\mathbb H$ , where ...
0
votes
1answer
136 views

Proof for ring isomorphism

Let $p$ be a prime, $\zeta$ a primitive $p$-th root of unity and $\Phi\in\mathbb{Z}[X]$ the $p$-th cyclotomic polynomial (i.e. $\Phi = 1 + X + X^2 + \ldots + X^{p-1}$). Let further be ...
5
votes
2answers
283 views

Prove that $p$ is prime in $\mathbb{Z}[\sqrt{-3}]$ if and only if $x^2+3$ is irreducible in $\mathbb{F}_p[x]$.

I'm having trouble with this particular homework problem. I think I have one direction: Let $R=\mathbb{Z}[\sqrt{-3}]$. If $p$ is prime, then $p$ is irreducible in $R$, since $R$ is an integral ...
6
votes
2answers
1k views

Why is the localization at a prime ideal a local ring?

I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is ...
3
votes
1answer
44 views

How does one interpret the factor ring of a factor ring?

If I have $\phi: R/I \rightarrow A$ as a ring map, how can I interpret $(R/I)/\ker(\phi) \cong A$? It seems pretty easy, but I just can't wrap my brain around it this morning.
27
votes
3answers
797 views

Ideals of $\mathbb{Z}[X]$

Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ...
4
votes
2answers
95 views

Commutativity in a Unital Banach Algebra

Let $ A $ be a unital Banach algebra and $ S $ a non-empty subset of $ A $. The centralizer of $ S $ is defined as $$ Z(S) \stackrel{\text{def}}{=} \{ a \in A ~|~ \text{$ as = sa $ for all $ s \in S ...
3
votes
2answers
157 views

How to think about quotient rings

I'm having a hard time wrapping my head around quotient rings that isn't something of the form $\mathbb{Z}/n\mathbb{Z}$. Is it correct to think of $\mathbb{Z}[x]/(x-1)$ as the ring of functions ...
2
votes
1answer
69 views

How many solutions to prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$

Let $a,b,c$ be integers, no sign restriction. Let $p$ be a given prime. How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ? Note, from Heron's ...
4
votes
1answer
68 views

Sums of powers of prime ideals in a Dedekind domain.

Let $P$ and $Q$ be distinct nonzero prime ideals of a Dedekind domain, $R$. Show that $P^{m} + Q^{n} = R$ for integers $m$, $n$. It's clear to me that $P+Q=R$ since both $P$ and $Q$ are maximal ...
17
votes
3answers
562 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
3
votes
3answers
154 views

How many solutions to prime = $a^3+b^3+c^3 - 3abc$

Let $a,b,c$ be integers. Let $p$ be a given prime. How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ? Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of ...
-1
votes
1answer
193 views

How to turn a group ring $R(G)$ into a ring?

Let $R(G)$ be a given abelian group ring. Any abelian group ring is isomorphic to an abelian ring. I know how to express (isomorphism) some group rings as a ring. But I wonder if there is a general ...
5
votes
2answers
123 views

Irreducibility of $x^2+x+4\in {\Bbb Z}_p[x]$ over ${\Bbb Z}_p$?

The following is an exercise in abstract algebra. Show that $x^2+x+4$ is irreducible over ${\Bbb Z}_{11}$. One test all the elements in ${\Bbb Z}_{11}$ to show that $x^2+x+4$ has no zeros in ...
1
vote
2answers
92 views

How many solutions to prime = $(d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$?

Let $a,b,c,d$ be integers $>-1$. Let $p$ be a given prime. How to find the number of solutions to $p = (d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$ ? I assumed that this polynomial above does not ...