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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3
votes
2answers
389 views

Example of non-Noetherian non-UFD Krull domain?

After a confusing session of hopping through Wikipedia articles, I started trying to summarize for myself some of the inclusions and relations among the many types of integral domains. Right now I'm ...
-2
votes
1answer
109 views

$End_{\mathbb{C}} ( \mathbb{C}[x])$ and Weyl algebra

How do you see this? $End_{\mathbb{C}} ( \mathbb{C}[x])$ As $M_{n}(\mathbb{C})=End_{\mathbb{C}} ( \mathbb{C}[x])$, so it just a matric with basis of polyonial? Take the weyl algebra $A_1=\{ ...
5
votes
2answers
780 views

A question about the nilradical

I've been thinking about the nilradical and I am wondering if the nilradical is the smallest, non-zero ideal of the ring. The reason why I'm asking is the following: Every ideal contains $0$. If $x ...
1
vote
1answer
102 views

Proof of property of local rings

I would like to prove: If every $x \in R - m$ where $R$ is a ring and $m$ is an ideal is a unit then $R$ is local with maximal ideal $m$ Can you tell me if my proof is right: Want to show that ...
4
votes
3answers
166 views

Involutions on commutative rings

I found that all the commutative rings with involution I know are the following: complex number with complex conjugation (plus similar constructions based on rationals and its extensions), any ...
8
votes
4answers
818 views

Explicit examples of infinitely many irreducible polynomials in k[x]

My question is the following. Is it possible to give examples of infinitely many irreducible polynomials in a polynomial ring $k[x]$ with $k$ a field? I'm interested in this because I'm ...
5
votes
2answers
608 views

Is the determinant of a zero divisor zero?

Suppose that $A$ is a zero divisor in the ring of $(n\times n)$-matrices over the ring $R$. Is $\det(A) =0$ if $R$ is a field? Is $\det(A) =0$ if $R$ is an integral domain? It's not necessarily ...
5
votes
1answer
150 views

Only proper ideal is $\{0\}$ $\implies f:A \rightarrow B$ is injective

I'm thinking about the proof of the following: If $A,B$ are rings and the only proper ideal of $A$ is $\{0\}$ and $f:A \rightarrow B$ is a ring homomorphism then $f$ is injective. My proof: Assume ...
2
votes
2answers
827 views

Bijection between ideals of $R/I$ and ideals containing $I$

I read that there is a one-one correspondence between the ideals of $R/I$ and the ideals containing $I$. ($R$ is a ring and $I$ is any ideal in $R$) Is this bijection obvious? It's not to me. Can ...
2
votes
1answer
767 views

The necessary and sufficient condition for a unit element in Euclidean Domain

I am trying to prove that in Euclidean domain D with Euclidean function d, u in D is a unit if and only if d(u)=d(1). Suppose u is a unit, then there exist v in D such that uv=1, this implies u\1 so ...
5
votes
2answers
1k views

Any prime is irreducible

I have seen many proofs about a prime element is irreducible, but up to now I am thinking whether this result is true for any ring. Recently, I got this proof: Suppose that $a$ is prime,and that $a = ...
3
votes
0answers
70 views

The number of local rings $R$ such that $R^\ast$ is cyclic of order $n$

For $n>0$, let $c_n$ be the number of local rings $R$ such that $R^\ast$ is cyclic of order $n$. Note that $c_1 =1$. (A local ring $R$ such that $R^\ast = \{1\}$ has precisely two elements. See Is ...
2
votes
3answers
701 views

Integral domain that is not a division ring

What is an example of integral domain that is not a division ring?
2
votes
4answers
127 views

The valuation ring $R$ in $K(T)$, such that $K[T] \subsetneq R \subsetneq K(T)$

$K$ is an algebraically closed field, $K[T]$ is the ring of polynomials of one indeterminate over $K$, and $K(T)$ is its field of fractions. A valuation ring $R$ in $K(T)$ which includes $k[T]$ and ...
6
votes
4answers
2k views

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
2
votes
2answers
157 views

Proving a function is a ring homomorphism

If $R$ is an integral domain with char $p$ where $p>0$ and $f:R\to R$ where $f(x)=x^p$ How would one go about showing addition is preserved? e.g. $f(a+b)=f(a)+f(b)$? Multiplication is obvious. So ...
1
vote
5answers
315 views

Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$?

Let $R$ be a commutative ring with 1, we define $$N(R):=\{ a\in R \mid \exists k\in \mathbb{N}:a^k=0\}$$ and $$U(R):=\{ a\in R \mid a\mbox{ is invertible} \}.$$ Could anyone help me prove that if ...
3
votes
4answers
185 views

How to prove $\bar{m}$ is a zero divisor in $\mathbb{Z}_n$ if and only if $m,n$ are not coprime

Let us consider the ring $\mathbb{Z}_n$ where $\bar{m}\in\mathbb{Z}_n$ Could anyone help me prove that $\bar{m}$ is a zero divisor in $\mathbb{Z}_n$ if and only if $m,n$ are not coprime So far I ...
2
votes
1answer
123 views

How to get the general form of functions in the ring of trigonometric polynomials

The ring of trigonometric functions over $\mathbb{R}$ is the ring generated by $\sin{x}$ and $\cos{x}$. What's the reason for why any function $f$ in this ring can be written as $$ ...
6
votes
1answer
227 views

Why are the centers of $R$ and $\text{End}_{\text{Ab}}(R)$ isomorphic?

Let $R$ be a ring (with unity) and let $E = \text{End}_{\text{Ab}}(R)$ be the ring of endomorphisms of $R$'s underlying abelian group. There is an injective ring homomorphism $\lambda: R \to E$ given ...
0
votes
1answer
79 views

If $B$ is a finite boolean alebgra and $a_1,\ldots,a_k$ are the atoms of $B$: $\forall i$ $a_ix=a_i x$, why is $x=a_1+\ldots +a_k$

Let $B$ be a finite boolean algebra. Define for $a,b\in B$ $a\leq b$ if $ab=a$ If $x\in B$ and $a_1,\dots,a_k$ are the atoms of B (e.g. $a\neq 0$ and if $b\in B$ such that $0\leq b \leq a$ then ...
4
votes
1answer
315 views

Does this “extension property” for polynomial rings satisfy a universal property?

On page 151 of Paolo Aluffi's Algebra: Chapter 0, an important property of the polynomial ring $\mathbb{Z}[x_1, \cdots, x_n]$ is introduced, namely that it's initial in the category of set functions ...
2
votes
1answer
137 views

Is the functor $\mbox{Rings}\rightarrow \mbox{Sets}$ given by $R \mapsto \{\pm 1 \in R\}$ corepresentable?

Is the function $\mbox{Rings}\rightarrow\mbox{Sets}$ given by $R\mapsto \{\pm 1\in R\}$ corepresentable? Of course this might be problematic in characteristic 2 since this set is then a singleton, ...
1
vote
1answer
273 views

Indecomposable rings with nontrivial idempotents

I am looking for examples of indecomposable rings with nontrivial idempotents. The only examples I can think of are matrix rings. Are there other examples?
3
votes
3answers
175 views

Prove that $(I-T)(I+T)^{-1}$ is an involution

I have to prove that if $V$ is a finite-dimensional vector space over a field of characteristic not 2, and $T$ is an endomorphism such that $\det(I+T) \neq 0$ then $T \mapsto (I-T)(I+T)^{-1}$ is an ...
1
vote
1answer
457 views

Nilpotent elements of noncommutative ring do not form ideal

In a commutative ring, the nilpotent elements form an ideal called the nilradical. The proof that the nilradical is an ideal uses the binomial theorem, which doesn't hold in noncommutative rings. Is ...
6
votes
1answer
235 views

Is there a classification of local rings with trivial group of units?

Out of curiosity, is there a classification of all local rings with trivial group of units? I suppose what I'm trying to ask is, if I asked for all local rings $R$ with $R^\times=\{1\}$, what would ...
1
vote
2answers
556 views

When is the preimage of prime ideal is not a prime ideal?

If $f\colon R\to S$ is a ring homomorphism such that $f(1)=1$, it's straightforward to show that the preimage of a prime ideal is again a prime ideal. What happens though if $f(1)\neq 1$? I use the ...
0
votes
3answers
210 views

Finite commutative domain implies field proof [duplicate]

Possible Duplicate: Why is a finite integral domain always field? How do you prove this; Let R be a finite commutative domain. Prove that R is a field. I need to know this because doing a ...
2
votes
1answer
332 views

Multiplicative monoid of a commutative ring

Is there any good description of the multiplicative monoid of a commutative ring in general? Or in special cases? I understand that in a UFD, it is the result of adjoining a zero to the Cartesian ...
1
vote
0answers
70 views

A module with a certain property

I know that the definition of reflexive module is that the $R$-module $M$ should be isomomorphic to its double dual $M^{**}$ via the canonical map $M\rightarrow M^{**}$. I'd like to know an example of ...
3
votes
1answer
81 views

If $x \in \mathbb{Z}/p^n\mathbb{Z}$, then $x = \mbox{unit} \times p^e$?

Let $R = \mathbb{Z}/p^n\mathbb{Z}$ where $p$ is a prime, and $n \ge 1.$ Let $\mathcal{U}(R)$ denote the units of $R.$ Is it possible to write any element $x \in R$ as $$x = up^e$$ where $u \in ...
0
votes
1answer
204 views

Torsion submodule

I'm looking for examples of a commutative ring $R$ such that its set of torsion elements $T$ is a submodule of $R$ (seen as an $R$-module), but such that $R/T$ is not torsion free. Anyone?
1
vote
1answer
303 views

Showing a Ring of endomorphisms is isomorphic to a Ring

Im trying to show that $\mathrm{End}(\langle \mathbb{Z},+\rangle)$ is naturally isomorphic to $\langle \mathbb{Z},+,\cdot\rangle$, but I'm not quite sure which ring homomorphism to use. Thank you
3
votes
1answer
101 views

Counting bases to which numbers are pseudoprime

Let $n=p_1^{a_1}\cdots p_k^{a_k}$ be an odd composite. Then the number of bases $1\le b\le n-1$ for which n is a strong pseudoprime is $$ \left(1 + \frac{2^{k\nu}-1}{2^k-1}\right) ...
4
votes
4answers
862 views

Modular Arithmetic question, possibly involving Chinese remainder theorem

'6 professors begin courses of lectures on Monday, Tuesday, Wednesday, Thursday, Friday and Saturday, and announce their intentions of lecturing at intervals of 2,3,4,1,6,5 days respectively. The ...
14
votes
2answers
824 views

Can a nonzero polynomial evaluate to the zero function in a suitable infinite ring of char 0?

I shall assume all rings to be commutative in this question. The impatient can scroll down to the "blockquote" to read the actual question. Whenever we have a polynomial over a ring, it defines a ...
3
votes
2answers
353 views

Is the algebraic norm of an euclidean integer ring is also an euclidean domain norm?

Let K be a finite extension of $\mathbb{Q}$ (a number field) and $\mathcal{O}_K$ its ring of integers. One defines the norm of an element $\alpha\in K$ to be the determinant of the transformation ...
22
votes
1answer
628 views

$Ra=Rb$ if and only if $aR=bR$

On which classes of (non commutative) rings we have the following property: $aR=bR$ if and only if $Ra=Rb$ ? While I googling around I found the notion of "Duo Ring" in which $aR=Ra$ for every $a\in ...
1
vote
1answer
331 views

Non-principal prime ideals of $\mathbb{Z}[x]$

How can you show that the non-principal prime ideals of $\mathbb{Z}[x]$ can be generated by only two elements, a prime number $p$ and an irreducible polynomial not in $p\mathbb{Z}[x]$? I can get to ...
7
votes
1answer
206 views

invertibility of elements in a Noetherian ring

Let $A$ be a left Noetherian ring. How do I show that every element $a\in A$ which is left invertible is actually two-sided invertible?
8
votes
2answers
392 views

Is this ring Noetherian?

The subring of $\mathbb{C}[x,y]$ consisting of all polynomials $f(x,y)$ whose gradient vanishes at the point $x=y=0$. Is this ring Noetherian?
1
vote
1answer
614 views

sub-module of a direct sum of modules that is not the direct sum of submodules

Let M and N be left R-Modules, is it possible to construct an example of a sub-module of $M \oplus N$ that is not a direct sum of a submodule of M and a submodule of N? I don't know a whole lot of ...
4
votes
2answers
152 views

Subrings of formal series rings

Let $k$ be a field and $A = k[[x_1, \dots, x_n ]]$ be the ring of formal series in $n$ variables. Consider $g_1, \dots, g_m \in A$ such that $g_1(0) = \cdots = g_m(0) = 0$. For every $f \in k[[t_1, ...
1
vote
0answers
163 views

Unity in a partial ring of quotients

I have the following question If we let $R$ be a commutative ring and we let $T$ be a nonempty subset of $R$ closed under multiplication not containing zero nor divisors of zero, i am asked to show ...
11
votes
6answers
2k views

Why is a finite integral domain always field?

This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse: let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$, then as $R$ is closed under ...
2
votes
1answer
230 views

If $R$ is a subring of $\mathbb{Q}$, $m/n\in R$ with $\gcd(m,n)=1$, and $p$ a prime factor of $n$, is $1/p$ an element of $R$?

Let $R$ be any subring (with $1$) of $\mathbb{Q}$. Let $m/n$ be an element of $R$ where $m$ and $n$ are coprime. Let $p$ be a prime divisor of $n$. Could anyone help me show that $1/p$ is an element ...
5
votes
1answer
483 views

Irreducible quadratics in polynomial ring of two variables over algebraically closed field

I'm currently stuck at problem 1.1 c) in Hartshorne's algebraic geometry book. I just can't let it go. Setting is as title says (field $k$, variables $x$ and $y$). Problem 1.1. a) and b) concerns ...
3
votes
4answers
522 views

Are there any nontrivial, finite subrings of an infinite ring?

For example, $S\subset\mathbb{R}$ where $S=\{0\}$ is the trivial subring which is finite. Is there a nontrivial subring of an infinite ring (i.e. of $\mathbb{R}$ or not) that is non-infinite? This ...
11
votes
4answers
2k views

Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.