This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (2)

1
vote
1answer
36 views

Show that a polynomial f(x) over a field k is irreducible if and only if the polynomial f(x + 1) is irreducible.

I am very much unsure what definitions and formulas are relevant for this question. I've toyed around with the lemma "An element a ∈ R is a root of a polynomial f ∈ R[x] if and only if (x − a) divides ...
0
votes
2answers
68 views

Prove that $I$ is a maximal ideal

I have a question. To show that the ideal $I=\langle f(x)\rangle $ is a maximal ideal of $K[x]$ do I have to show that $f(x)$ is irreducible in $K[x]$? Or is there an other way to prove that $I$ is a ...
0
votes
1answer
59 views

Given $x$ and $y$ in $\mathbb{Z}[i]$, find $q$ and $r$ such that $x=qy+r$.

Find $q, r \in \mathbb{Z}[i]$ such that: $1 + 5i = (1 + 2i)q + r$ with $|r| < 2$, $1 + 5i = (2i)q + r$ with $|r| < 2$. My only train of thought is that $r = 1+0i$, $0+i$ or ...
2
votes
1answer
113 views

On Bounded Index of Nilpotency of $R[x]$ and $M_n(R)$

A ring $R$ is said to have a bounded index (of nilpotency) if there is a positive integer $n$ such that $x^n=0$ for every nilpotent $x∈R$. Can anyone give me an example of a ring $R$ which has a ...
1
vote
1answer
30 views

When is $r = r^{2}$ in $\mathbb{Z}/p^{l}\mathbb{Z}$?

When is $r = r^{2}$ in $\mathbb{Z}/p^{l}\mathbb{Z}$, where p is a prime number and l is a natural number? It obviously is the case for [0] and [1], but I am having difficulties proving that it's not ...
2
votes
3answers
87 views

When does a ring map $R\to S$ produce a group epimorphism $GL_n(R)\to GL_n(S)$?

Let $R$ and $S$ be rings with $1$ (not necessarily commutative) and $f:R\to S$ a ring homomorphism preserving $1$. Let $\bar{f}$ be the ring map $M_n(R)\to M_n(S)$ given by $f$ acting on the matrix ...
0
votes
0answers
20 views

Why $f(0)\neq 0$ where $f $ is a polynomial over the field $F_q$ and $deg(f)=m > 0$?

To construct the Residue class ring $F_q[x]/(f)$ having $q^m-1$ non-zero elements. Is it necessary for $f(0) \neq 0$? Why or why not? I have worked with different examples such as $x^3+x=f \in ...
1
vote
0answers
77 views

Does a ring map $f:R\to S$ induce a homomorphism $GL_n(R)\to GL_n(S)$?

Let $R$ and $S$ be commutative rings with $1$ and $f:R\to S$ a ring homomorphism. Does $f$ induce a group homomorphism $GL_n(R) \to GL_n(S)$? Progress I first consider the map $\bar{f}:M_n(R)\to ...
4
votes
2answers
93 views

Describing the ring $\mathbb{Z}[x]/(x-8,2x-6)$

How can one describe the ring $\mathbb{Z}[x]/(x-8,2x-6)$? What is meant by describing it? How does an element of the ideal generated by $x-8$ and $2x-6$ look like?
4
votes
1answer
47 views

If a polynomial ring has a zero divisor, then the ring has a zero divisor

This was asked here: Zero divisor in $R[x]$ But what I'm really asking is: If $f(x)=a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, isn't every $a_i$ a zero divisor in $R$? Let ...
0
votes
3answers
64 views

Prove that in a ring with at least two elements $0\neq 1$. [closed]

Let R be a non-trivial ring then prove $0\neq 1$.
0
votes
1answer
51 views

What are the $2$-dimensional algebras over any arbitrary field?

As a follow-up of this question, I would like to ask, what are the $2$-dimensional algebras over $\mathbb R$, $\mathbb Q$, or any arbitrary field? Can we classify them?
4
votes
2answers
95 views

Product of ideals for Nakayama's Lemma

The result to be proved is the following: Let $R$ be a local Noetherian ring. Then the minimum number of generators of the unique maximal ideal $P$ equals the dimension of $P/P^2$ as a vector space ...
1
vote
2answers
49 views

In a $\mathbb{Z}$-graded ring with unity, $1$ is homogeneous of degree zero

Suppose I have a $\mathbb{Z}$-graded ring (commutative) $A$ with unity. I am sure that the unity $1 \in A$ has degree $0$. I was wondering how could one show that? (I am guessing we don't have ...
0
votes
1answer
88 views

How to show non unit $x$ is idempotent in $R$ if $xR+aR=R$ for all $a\in R\smallsetminus (J(R)\cup U(R)\cup\{x\})$?

Let $R$ be a commutative ring with the multiplicative identity. Let $x$ be a non unit element of $R$ such that $xR+aR=R$ for all $a\in R\smallsetminus (J(R)\cup U(R)\cup\{x\})$ where $U(R)$ is the ...
4
votes
1answer
109 views

Which rings containing the complex field are, as vector spaces over that field, isomorphic to $\mathbb{C}^2$?

Which rings $R$ containing (as a subring) the complex field $\mathbb{C}$ are, as a vector space over that field, isomorphic to $\mathbb{C}^2$? In other words: what are the two-dimensional unital ...
5
votes
1answer
42 views

In a Euclidean ring, could we prove irreducible implies prime directly?

In a UFD, primeness and irreducibility are equivalent. In particular, every Euclidean ring which is an integral domain is a UFD. My question is this: is it possible to prove that "irreducible ...
4
votes
1answer
136 views

Are all simple left modules over a simple left artinian ring isomorphic?

I've a basic question. $R$ is a simple left artinian ring. I want to show that all simple left $R$-modules are isomorphic. A simple $R$-module $M$ is isomorphic to $R/J$ where $J$ is a maximal ...
3
votes
1answer
55 views

An example of a function failing to be a ring homomorphism

Can anyone give an example of two rings $R$ and $S$ and a function $f$:$R$ → $S$ which preserves multiplication and addition but with $f$($1_R$) $\neq$ $1_S$. Thus $f$ failing to be a ring ...
2
votes
2answers
459 views

Commutative ring is semisimple iff it's isomorphic to a finite direct product of fields.

I am trying to prove the following: Let $R$ be a commutative ring. Prove that $R$ is semisimple if and only if it is isomorphic to a direct product of a finite number of fields. Suppose $R$ is a ...
2
votes
1answer
97 views

Multiplicity of a dual simple module in the dual module?

Let $A$ be a finite dimensional $k$ algebra. Let $S$ be a simple left $A$-module and $M$ be any left $A$ module. Then my first question is that is it true $$[M:S]=[M^*:S^*]$$ where ...
3
votes
3answers
427 views

Can a principal ideal contain a non-principal ideal?

I am trying to show that if $R$ is an integral domain such that every prime ideal of $R$ is principal then every ideal of $R$ is principal. To start this, suppose that $P$ is the set of ideals of ...
2
votes
0answers
30 views

Is this problem still open:If G is a torsion free group and F is a field then group ring F[G] is an intergral domain.?

I know this question has answer for when G is infinite cyclic group group.Does there is a general proof? Could anyone give me some references...
2
votes
1answer
54 views

Reducible in $R[X]$ implies reducibility in $(R/I)[X]$

Let $R$ be an integral domain, $I$ a proper ideal of $R$; $\pi:R \to R/I$ the canonical projection. Let $f=\sum_{i=0}^n a_iX^i$ be a monic polynomial and $\overline{f}=\sum_{i=0}^n \pi(a_i)X^i \in ...
0
votes
1answer
43 views

A field with four elements

Determine the additive group of the field of four elements. My attempt:Consider $(F,+,.) $ the field of four elements.Now $0,1\in F$ as $(F,+,.) $ is a field .As it contains $4$ elements $\exists ...
3
votes
2answers
172 views

Problem related to the ring $\mathbb Z[\sqrt{d}]$

Let $d \in \mathbb Z$ and let $\sqrt{d} \in \mathbb C$ be a square root of $d$. Consider the subring of $\mathbb C$, $$\mathbb Z[{d}]=\{a+b\sqrt{d}: a,b \in \mathbb Z \}$$ and we define the norm of an ...
2
votes
1answer
70 views

Finding an isomorphism between $\mathbb{Z}[i]/(7)$ and $\mathbb{Z}[\sqrt{-2}]/(7)$

I was having some trouble finding an explicit isomorphism between $\mathbb{Z}[i]/(7)$ and $\mathbb{Z}[\sqrt{-2}]/(7)$. $\textbf{What I have noticed is}:$ 7 is a prime element in $\mathbb{Z}[i]$ so ...
0
votes
3answers
29 views

Convergence in ordered rings

The question is, essentially, whether we can compute the limits of some sequences in ordered rings. Given an ordered ring $R$ with ordering $\le$, we can say that a sequence $(a_n) \to a$ if and only ...
3
votes
1answer
93 views

Finding simple modules in fields

I am struggling with finding all simple are modules for general rings i know that the simple modules of R correspond with the simple modules of R/rad(R), where rad(R) is the Jacobson radical but i ...
1
vote
5answers
58 views

Isomorphism of quotient fields

Consider $\mathbb R $ and $\mathbb Q$ with usual meanings.Which of the following rings are isomorphic? a. $\mathbb Q[x]/\langle x^2+1\rangle $ and $\mathbb Q[x]/\langle x^2+x+1\rangle$ b. $\mathbb ...
1
vote
1answer
52 views

What do elements of this quotient ring look like and why?

Let $R = \{a+b \sqrt 2| a,b $ integers$\}$. Let $M = \{a+b \sqrt 2| a,b $ integers and $5|a $ and $ 5|b\}$ be its ideal. How would you write out $R/M$ in this form?
2
votes
1answer
78 views

Proving $M$ is maximal if the quotient ring $R/M$ is a field.

Let $R$ be a ring with unit element and ideal, $M$, such that $R/M$ is a field. Prove $M$ is maximal ideal. I know that because $R/M$ is a field, its only ideals are $(0)$ and itself. Also, I ...
3
votes
3answers
205 views

Basic Ring Theory Question involving the unit element

If $ \phi: R \to R'$ is a homomorphism of $R$ onto $R'$ and $R$ has a unit element, $1$, show that $\phi(1)$ is the unit element of $R'$. I am having trouble proving this. I think I just need a hint ...
1
vote
1answer
82 views

Question concerning the chinese remainder theorem for commutative rings

let $S$ be a commutative ring and $I_1,...,I_n\unlhd S$, such that $I_i+I_j=S\ \forall i\neq j$. Let $g_1,...,g_n\in S$. Why are there $h_1,...,h_n,h'\in S$, such that ...
1
vote
0answers
38 views

Radical of an ideal in a finitely generated ring over $k$ is the intersection of maximal ideals containing it. [duplicate]

From Matsumura p.34 Let $k$ be a field, $A$ a ring which is finitely generated over $k$, and $I$ a proper ideal of $A$; then the radical of $I$ is the intersection of all maximal ideals containing ...
2
votes
1answer
29 views

2 questions concerning identities of closed subspaces of $spec(S)$ for a commutative ring $S$

I have the following questions: Let $S$ be a commutative ring and let $M,N$ be closed subspaces of $spec(S)$, such that $M\cap N=\emptyset$. 1) Why are there ideals $I_1,I_2\unlhd S$, such that ...
4
votes
0answers
71 views

Prove $\sqrt{-7} \not\in \mathbb{Z}\left[\frac{2+3\sqrt{-7}}{4}\right]$

I have the following problem. Consider the ring $\mathbb{Z}$ and define: $$x = \sqrt{-7}\qquad z = \frac{2+3x}{4}$$ Show that $\mathbb{Z}[x] \not\subset \mathbb{Z}[z]$ and $\mathbb{Z}[z] ...
6
votes
2answers
247 views

Product of Nilpotent ideal and simple module

I am stuck with trying to show that if an ideal I of a ring R is Nilpotent and M is a Simple R-Module, then IM = 0 I have attempted showing this by using the fact that the annihilator of a simple ...
1
vote
2answers
77 views

Understanding $\mathbb{Q}[x]/(x^{2}+1)^{2}$ contains…..

This is a question about an answer to this question: Is $\mathbb{Z}[x] / \langle (x^2 + 1)^2 \rangle$ isomorphic to a familiar ring? Here the answer says that the ring $\mathbb{Q}[x]/(x^{2}+1)^{2}$ ...
1
vote
0answers
38 views

The ring $\mathbb{Z}_2 \times \mathbb{Z}_2$ is a domain [duplicate]

True\False :The ring $\mathbb{Z}_2 \times \mathbb{Z}_2$ is a domain solution True A commutative ring with identity is said to be an integral domain if it has no zero divisors.
2
votes
1answer
93 views

Ring structures on abelian groups

My question is: given an abelian group $G$ with addition $+$, is there some natural multiplicative structure that arises so that we can define a ring $(G, +, \cdot)$. For instance, multiplication on ...
-1
votes
1answer
78 views

Finding a coordinate ring

I am having hard time in calculating (or constructing) $\displaystyle\frac{\mathbb C[x,y]}{\langle y^2 - x^3 - x\rangle}$. I tried homogenizing the ideal $y^2 - x^3 -x $ to $ wy^2 - x^3 - xw^2$. But ...
1
vote
0answers
56 views

Universal property of the Tensor Algebra

Let M be an A-module over a commutative ring A. For any A-algebra N and A-module homomorphism $\phi : M \rightarrow N$ there is a unique A-algebra homomorphism $\Phi : T(M) \rightarrow N$ (where T(M) ...
1
vote
0answers
27 views

Modules over PIDs

Let $M=\mathbb{Z}^4/N$ where $N$ is a subgroup of $\mathbb{Z}^4$ generated by $(1,0,-1,3)$ and $(2,4,8,-6)$. Recognize $M$ as a product of cyclic groups. Here I have to use the following Theorem: If ...
1
vote
1answer
112 views

A simple module is necessarily the socle of its injective hull?

According to the wikipedia page on injective hulls "a simple module is necessarily the socle of its injective hull". It is clear to me that why a simple module is a subset to the socle of of its ...
2
votes
1answer
42 views

Intersection of nonzero ideals in a right Noetherian domain is nonzero

I've been asked to show that in a right Noetherian domain, the intersection of nonzero right ideals is nonzero. A hint is given, saying that if not, then any nonzero right ideal contains a direct sum ...
0
votes
1answer
40 views

Powers of complexes modulo a prime $p$

We have, for a residue number system, $a^{n+(p-1)} \equiv a^n \bmod p$. In other words, the powers of $a$ repeat after $p-1$ iterations. We can work with complex numbers by representing a number $$n ...
0
votes
2answers
267 views

Polynomial rings- multiplicative inverse

I need to solve the following question in ring theory. Show that $(Q[x])/\langle{x^2+x+4}\rangle$ is a field. To show that $(Q[x])/\langle{x^2+x+4}\rangle$ is a field, the only thing I need to do ...
3
votes
1answer
312 views

Example of a commutative ring which is not a subring of a commutative ring where every non-invertible element is a zero-divisor

I don't remember whether there was a special name for a commutative ring where every non-invertible element is a zero-divisor. And I also forgot the different ways in which a non-invertible element ...
2
votes
3answers
65 views

Are these two rings isomorphic?

I have these two rings $k[x,y,1/y]/(x^2 +1 - y^2)$ and $k[u,v,1/v]/ (u^2 + v^2 -1)$, where $k$ is a field, and I was wondering if these two rings were isomorphic or not. I would greatly ...