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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11
votes
2answers
703 views

Number of elements in the quotient ring $\mathbb{Z}[X]/(X^2-3, 2X+4)$

I had to calculate the number of elements of this quotient ring: $$R = \mathbb{Z}[X]/(X^2-3, 2X+4).$$ This is what I've got by myself and by using an internet source: Writing the ring $R = ...
1
vote
2answers
163 views

Sum of ideals in the polynomial ring

Could someone explain to me how to find a sum of ideals where $I=(x+y)$ and $J=(x)$? The answer to this is $I+J=(x,y)$ and we work in the polynomial ring $k[x,y]/(xy)$. I know that the definition ...
1
vote
1answer
49 views

Ideals of nested PID's

Let $R\subset K$ be principal ideal domains. If $a,b$ are nonzero elements of $R$, prove that $I=J\cap R$, where $I$ and $J$ denote the ideals generated by $a,b$ in $R$ and $K$, respectively. Showing ...
2
votes
2answers
125 views

Generalizing norms for modules

An normed space is defined as a vector space V plus a norm operation over $V$. Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields? What I'm ...
9
votes
2answers
314 views

Cosets modulo $(2+i)$ in $\mathbb{Z}[i]$

I've been trying to solve this problem but I got stuck on it: Given is the ideal $I=(2+i)$ in the ring of Gaussian integers $\mathbb{Z}[i]$. How many elements does the quotient ring ...
3
votes
2answers
245 views

Example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable ...
3
votes
1answer
69 views

proving isomorphism of two $k$-algebras

Let $k$ be a field. I would like to prove that $k[x,y]/(x^3-y^2) \cong k[t^2,t^3]$. Of course, intuitively, i can readily see that this must be the case. More formally, i define a homomorphism ...
3
votes
4answers
249 views

Let $R$ be a ring that has no nonzero nilpotent commutators. If $e\in R$ is an idempotent, then $e\in Z(R)$.

Let $R$ be a ring that has no nonzero nilpotent commutators. If $e\in R$ is an idempotent, then $e\in Z(R)$. I have a problem with proving this theorem. I don't know how to understand nonzero ...
4
votes
2answers
211 views

If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.

If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative. I need to proof this theorem and I have something like this below. However, I do not know how to continue this proof. ...
3
votes
3answers
342 views

Order of $\mathbb{Z}[i]/(1+i)$ [duplicate]

I have to calculate the order of the ring $\mathbb{Z}[i]/(1+i)$. This is how far I am: If $a+bi\in \mathbb{Z}[i]/(1+i)$ then there are $n,m\in \mathbb{Z}$ such that $a+bi\equiv 0+ni$ or $a+bi\equiv ...
9
votes
4answers
679 views

$\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$

I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$. I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be ...
0
votes
3answers
106 views

How to prove that a particular ideal is the kernel of a ring homomorphism?

I have to prove that $(1+3i)$ is the kernel of the homomorphism $f:\Bbb{Z}[i]\to \Bbb{Z}/10\Bbb{Z}$ defined by $f(a)\to a \mod 10, a\in \Bbb{Z}$, and $f(i)\to 3 \mod 10$. I know that $(1+3i)$ is ...
18
votes
1answer
545 views

When is a group ring an integral domain

If $R$ is an integral domain (I am having $\mathbb{Z}$ or a field in mind) and $G$ a (not necessarily finite) group then we can form the group ring $R(G)$. Note that if $g^{n+1} = e$ then ...
1
vote
2answers
128 views

Presentation of finite rings (fields)

One knows that every finite group is isomorphic to a subgroup of $\operatorname{GL}(n)$ for some $n$ large enough. Can every finite ring be represented by a ring of matrices, i.e., is every ring ...
5
votes
3answers
181 views

$(-a)^2=a^2$ in commutative ring?

Maybe this is a silly question, but how can I show that $(-a)^2=a^2$ in a commutative ring with $1$ for all $a$ in the ring? I know that $(-a)^2=(-a)\cdot(-a) =(-1)\cdot(-1)\cdot a^2$. So I ...
2
votes
2answers
394 views

What does Herstein mean by 'centroid of a ring'?

I'm currently reading Herstein's Noncommutative Rings, and the definition of the centroid of a ring is on page 46 of the book. Let $\text{End}(R)$ be the ring of endomorphisms of the additive group ...
2
votes
1answer
57 views

An advanced algebra question

How can I show that $\Bbb{R}(x)$ (the quotient field of $\Bbb{R}[x]$) is not a real closed field ?
0
votes
0answers
107 views

Non Classical Examples of Indecomposable Ideals

A classical example of a ring $R$ with an indecomposable ideal is the ring $C(X)$ of real valued continuous functions on $X$, where the $(0)$ ideal is not decomposable. Does anyone know other examples ...
1
vote
1answer
138 views

Direct sum of commutative rings

Let $R$ be a direct sum of ideals $R=R_1\oplus R_2\oplus\dots\oplus R_k$. Each ideal $R_i$ is commutative of order $p_{i}^{n_{i}}$ ($p$ is prime), and has a unity. How to show that the direct sum of ...
5
votes
1answer
92 views

Simple + Artinian = Semiprimitive

By a noncommutative ring I mean that it has no unit. I know that if some ring (say, $R$) is simple, then: $R^2 \neq (0)$ It only possesses $2$ two-sided ideals, namely $(0)$, and itself. And ...
9
votes
6answers
820 views

Examples of a commutative ring without an identity in which a maximal ideal is not a prime ideal

In a commutative ring with an identity, every maximal ideal is a prime ideal. However, if a commutative ring does not have an identity, I'm not sure this is true. I would like to know the ...
2
votes
4answers
158 views

Yet another characterization of the field $\mathbb{Z}/2\mathbb{Z}$

Let $R \neq 0$ be a ring which may not be commutative and may not have an identity. Suppose $R$ satisfies the following conditions. 1) $a^2 = a$ for every element $a$ of $R$. 2) $R$ has no two-sided ...
0
votes
1answer
186 views

Find all homomorphisms

Find all ring homomorphisms $\Phi$: $\mathbb{Z}_2 \rightarrow \mathbb{Z}_6$ and $\Phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2$.
2
votes
1answer
56 views

Properties of quasiregular elements in a matrix ring

I've been puzzling over one of the properties of quasiregular elements listed in the wikipedia article on the topic. An element of a ring $x$ is quasiregular (left, right) when $1-x$ has a ...
-1
votes
2answers
141 views

Ring of order $p^2$ and its characteristic

I suppose that this question might be very easy for some people. However, I have got problem to get it. Could anyone explain to me why the characteristic of a finite ring of order $p^2$ is $p$. I know ...
2
votes
0answers
59 views

If $\mathbb{C}[G]$ is Noetherian and $G$ has a representation on $V$, when must $V$ be finite-dimensional?

I know this is a bit vague, but please bare with me here. Let's assume that $G$ is a finitely-generated torsion group. I want to show that $G$ is a finite group if I add some conditions. I suspect ...
2
votes
1answer
97 views

Principal Ideal Groupring

Let $R[G]$ be a groupring (not necessarily commutative). Under which conditions on $R$ and $G$ is $R[G]$ a Principal Ideal Ring, respectively a Principal Ideal Domain (not commutative either)?
1
vote
2answers
116 views

is $I^2=I$ true?

Suppose $I$ is an ideal of a ring with $1$. I think that $II=I^2=I$ but I am stuck showing it. I can easily show that $I^2\subseteq I$, but I dont know how to show that $I\subseteq I^2$. So is it ...
3
votes
2answers
98 views

Difference between $R[c_1,c_2,\dots, c_n]$ and a finitely generated $R$-algebra.

What is the difference between $R[c_1,c_2,\dots, c_n]$ ($c_1, c_2,\dots, c_n\notin R$), where $R$ is a ring, and a finitely generated $R$-algebra? Is the difference that if $c_1, c_2,\dots, c_n$ ...
1
vote
1answer
206 views

$1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent [duplicate]

Let $(A, +, \cdot)$ be a ring with $1$. An element $a\in A$ is nilpotent if there exists $n\in \mathbb{N}$ so that $a^n=0$. Show that if $a$ is nilpotent then $1+a$ and $1-a$ are invertible.
2
votes
2answers
172 views

The Stone-Čech compactification of a space by the maximal ideals of the ring of bounded continuous functions from the space to $\mathbb{R}$

There is a claim that for any completely regular space, the maximal ideals of the ring of bounded continuous functions from $X$ to $\mathbb{R}$ forms the Stone-Čech compactification of $X$. How is the ...
1
vote
1answer
92 views

Noncommutative ring of order $np^2$

Could anyone help me to prove this theorem, please? Let $R_1$ be a ring of order $p^2$ which is the direct product of $C_p$ with itself and a minimal generating system for $R_1$ is $[(a,0),(0,a)]$, ...
2
votes
2answers
141 views

Prove $Z[\sqrt{d}] /(a + \sqrt{d}) \cong Z/nZ$

Let $a,d$ be integers with $d$ square free. Prove that $\mathbb{Z}[\sqrt{d}]/(a + \sqrt{d}) \cong \mathbb{Z}/n\mathbb{Z}$ where $n= |a^2- d|.$ I've tried attempting the problem by looking for a ...
2
votes
0answers
73 views

Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
2
votes
1answer
122 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
4
votes
5answers
359 views

Isomorphism of polynomial rings in several variables

I have been struggling with the following problem: How can one prove that if there is an isomorphism between several variable polynomial rings over a field $K$, $ \varphi : K[X_1, \dots, X_n] \to ...
1
vote
1answer
392 views

Proof of Hilbert's Nullstellensatz, weak form.

The statement of Hilbert's Nullstellensatz, weak form, as given here is "Let $f_1,f_2,\dots,f_n$ be polynomials in $K[x_1,x_2,\dots,x_n]$, where $K$ is an algebraically closed field. Then $1=\sum{g_t ...
1
vote
1answer
51 views

A question on generators

Suppose $I$ is an ideal in a ring $R$ which is finitely generated. Suppose on the other hand that there is some (possibly other) set of generators $\{g_t\colon t\in T\}\subset I$ which also generates ...
0
votes
1answer
80 views

An Isomorphism of Rings

Let $R$ be the ring of Quaternions over $\Bbb{Z}_{(3\Bbb{Z})}$ ($\Bbb{Z}$ localized in $3\Bbb{Z}$). Is it true that $\frac{R}{J(R)}$ can be represented as $M_2(\Bbb{Z}_3)$ ? ($\Bbb{Z}_3$ is the ...
0
votes
1answer
121 views

Proof of Noether's Normalization theorem.

As stated here, Noether's Normalization Theorem states: Suppose that $R$ is a finitely generated integral domain over a field $K$. Then there exists an algebraically independent subset ...
49
votes
12answers
5k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
-1
votes
1answer
99 views

If $P$ is a prime ideal of an integral domain $D$, then is $D$ equal to its localization at $P$?

I refer to this article on the localization of integral domains. Let $D$ be an integral domain, and $P$ a prime ideal of $D$. $$D_P=\{ab^{-1}\mid a\in D,b\notin P\}$$ Let us suppose $P\subset D$. ...
5
votes
2answers
147 views

characteristic prime or zero

Let $R$ be a ring with $1$ and without zero-divisors. I have to show that the characteristic of $R$ is a prime or zero. This is my attempt: This is equivalent to finding the kernel of the homomorphism ...
-1
votes
2answers
58 views

If a $k$-algebra is finitely generated, then does $k$ also have to be a finitely generated field?

Let $k$ be a field, and $A$ be a finitely generated $k$-algebra. Then does $k$ also have to be a finitely generated field? Motivation: Let $A$ be generated by $\{a_1,a_2,\dots,a_n\}$, and $k$ be ...
0
votes
0answers
46 views

in a finite ring, left inverse implies right inverse [duplicate]

Let $R$ be a finite ring with $1\neq 0$. Suppose there are $x,y\in R$ such that $xy=1$. I have to prove that this implies $yx=1$. I have read the question Left inverse implies right inverse in a ...
9
votes
2answers
206 views

Characterization of the field $\mathbb{Z}/2\mathbb{Z}$

Let $R \neq 0$ be a ring which may not be commutative and may not have an identity. Suppose $R$ satisfies the following conditions. 1) $a^2 = a$ for every element $a$ of $R$. 2) $ab \neq 0$ whenever ...
6
votes
3answers
514 views

Left Multiplication Ring Homomorphism

Assume we have a non-commutative unital ring $R$ and an element $r$ not in the center. Define a map $$\phi_r:R\rightarrow R$$ $$x\mapsto rx$$ Can this ever be a ring homomorphism? If it can be ...
1
vote
1answer
31 views

This ring with this metric can have just isosceles triangles

I'm trying to solve item (II) of this question: MY ATTEMPT Suppose $\rho(a,b)=2^{-m}$ and $\rho(a,c)=2^{-n}$ with $n\leq m$. Therefore, $a-b\in I^m\setminus I^{m+1}$ and $a-c\in I^n\setminus ...
0
votes
1answer
56 views

A homomorphism between two A-algebras.

A homomorphism between two algebras is described here. I want to describe a homomorphism $f:A[x_1,x_2,\dots,x_n]\to R$, where $R$ is an A-algebra. $A$ is a ring. Obviously, $A[x_1,x_2,\dots,x_n]$ is ...
0
votes
0answers
47 views

Ring structure on finite string of elements of a group

This is a reference request. Suppose $(G, \cdot)$ is a group and consider the structure on $G^{<\omega}$ where, for $\mathbf{p} = (p_1, \dots, p_n) \in G^{<\omega}$ and $\mathbf{q} = (q_1, ...