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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Easiest way to prove that a subset of even integers is closed under multiplication?

What's the easiest way of showing that; $2\mathbb{Z}\setminus (4n-2)\mathbb{Z}$ is closed under multiplication? (I'm trying to show that $(4n-2)$ is a prime element of $2\mathbb{Z}$ by showing ...
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4answers
1k views

Is the group of units of a finite ring cyclic?

The group of units of a finite field is cyclic. Is it true that the group of units of a finite ring is also cyclic? If not, where does the ring structure prevents us from obtaining the result that is ...
3
votes
1answer
763 views

Equivalence of definitions of prime ideal in commutative ring

Could someone prove that in a commutative ring $R$ the following two definitions of prime ideal are equivalent: 1) An ideal $P$ is prime if $P\neq R$ and if for all ideals $A,B$ of $R$ with ...
1
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1answer
138 views

Non-compatible ideals

Let $A$ be a complex unital algebra and let $M$ be a proper left ideal in $A$. Furthermore, suppose that $\{L_i\colon i \in I\}$ is an uncountable family of left-ideals such that 1) for each $i$ the ...
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2answers
275 views

Proving $R$ is a division ring

Theorem says that Suppose that $R$ is a domain that is an algebra over a subfield $k$. Assume that $R$ is finite dimensional $k$-vector space. Prove that $R$ is a division ring. I suppose should ...
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1answer
47 views

Help with factor groups and domains

I can't see it. $\Large{\frac{\mathbb{Z}[x]}{(x^2-29)}}$ I know you take general polynoimals with coefficients in $\mathbb{Z}$ and add $(x^2-29)$. However, I'm a bit confused on what it is. Like I ...
19
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2answers
2k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
4
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1answer
173 views

Why is a ring $R$ with the property that $r=r^2$ for each $r\in R$ so special?

The question is motivated by the following multiple-choice problem: If $R$ is a ring with the property that $r=r^2$ for each $r\in R$, which of the following must be true? I. $r+r=0$ for each ...
3
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1answer
188 views

Ring on an Elliptic Curve

I know that for a given elliptic curve $E$ we can define a group $G$ with the points on this curve. However, can we define a ring on it? That is, can we define a multiplication on the curve, where we ...
9
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1answer
821 views

Do these matrix rings have non-zero elements that are neither units nor zero divisors?

Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with entries in $R$. In addition, suppose that $R$ is a ring in which every non-zero element is ...
8
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1answer
926 views

Why is it called a 'ring', why is it called a 'field'?

The definitions of 'ring' and 'field' are pretty straightforward. For a ring (e.g. integers): addition is commutative $( 1 + 2 = 2 + 1 )$ addition and multiplication are associative $(2 +(2+2)) = ...
8
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1answer
778 views

What's an example of an ideal in $\mathbb{Z}[\sqrt{-n}]$ that is not principal?

Earlier I asked a question which showed that $\mathbb{Z}[\sqrt{-n}]$ for $n$ a square free integer greater than 3 is not a UFD. Since PID implies UFD, this also means $\mathbb{Z}[\sqrt{-n}]$ is not a ...
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1answer
3k views

Why is $\mathbb{Z}[\sqrt{-n}]$ not a UFD?

I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not an UFD. I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able ...
4
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2answers
977 views

A ring element with a left inverse but no right inverse?

Can I have a hint on how to construct a ring $A$ such that there are $a, b \in A$ for which $ab = 1$ but $ba \neq 1$, please? It seems that square matrices over a field are out of question because of ...
3
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0answers
62 views

Example of non-Krull integrally closed BFD?

Here's another question in the same spirit as my previous one: Are there any integrally closed BFDs which are not Krull domains? Some background information: A BFD (bounded factorization domain) is ...
3
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2answers
424 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 ...
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1answer
110 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
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2answers
882 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
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1answer
105 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
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3answers
177 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
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4answers
904 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
653 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
154 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 ...
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2answers
1k 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
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1answer
866 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 ...
6
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
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3answers
791 views

Integral domain that is not a division ring

What is an example of integral domain that is not a division ring?
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4answers
128 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 ...
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4answers
3k 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
162 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 ...
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5answers
350 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
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4answers
192 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
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1answer
132 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
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1answer
237 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
81 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 ...
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1answer
344 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
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1answer
139 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, ...
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1answer
319 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?
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3answers
176 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 ...
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1answer
530 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 ...
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1answer
244 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 ...
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2answers
590 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 ...
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3answers
238 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 ...
3
votes
2answers
379 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 ...
4
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1answer
91 views

A nonreflexive module isomorphic to its double dual

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 ...
3
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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
216 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?
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1answer
342 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
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1answer
106 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) ...