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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Equivalence of definitions of Noetherian Ring, another proof.

Let $R$ be a commutative ring with unity, then the following are equivalent -1. Every ideal in $R$ is finitely generated -3. Every nonempty collection of ideals of $R$ has a maximal element I will ...
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Showing that $|\text{Hom}_\mathbb{Q}(K,K)|=6$

Let $g\in \mathbb{Q}[X]$ be irreducible with $\text{deg}\;g=3$. We assume that only one of the roots of $g$ is in $\mathbb{R}$. So, $\alpha \in \mathbb{R}$. Let $L\subset\mathbb{C}$ be the splitting ...
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factoring numbers in $\mathbb{Z}[\sqrt{2}]$ into primes

How do I factor, say, 2 + 3$\sqrt{2}$ into primes in $\mathbb{Z}[\sqrt{2}]$? I know that primes are irreducible in $\mathbb{Z}[\sqrt{2}]$ and that units are of the form $\pm(1\pm\sqrt{2})^n$. How are ...
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Why does the singular locus of a random rational curve always have the same Hilbert function?

As background, let's first note that a random rational curve of degree $d$ in $\mathbb P^n$ is smooth when $n\ge 3$. Indeed, picking a random rational curve of degree $d$ in $\mathbb P^n$ corresponds ...
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Connection between Algebraic structures and Formal Systems

I am a college student with very little mathematics background (up to Calculus 152 at Rutgers University), but have become increasingly interested in computing and mathematics in the last year. I am ...
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(quasi)coherent rings for which $\dim R[T]\neq \dim R+1$

What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$? Why (hopefully geometrically) should we not always have equality? Notation. Let $I,J$ be two ideals of ...
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Motivating the Cayley-Dickson construction by proving Hurwitz's theorem

To me it seems the way to motivate the Cayley-Dickson construction is to prove Hurwitz's theorem, which is done over at Wikipedia. The theorem states the only real division algebras equipped with a ...
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$J$ maximal ideal of $A$ $\iff $ $A/J$ is field

I'm wondering where did this complicated proof that $J$ is a maximal ideal $\iff$ $A/J$ is a field. Is there an easy to look case where we can clearly see that when we take the quotient of the ring ...
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Monic Factors in $\mathbb Q[x]$ of a Monic $f \in \mathbb Z[x]$ are also in $\mathbb Z[x]$

The claim is that if $f \in \mathbb Z[x]$ is monic and has a factorization, say $f = pq \in \mathbb Q[x]$, such that $p, q$ are also monic, then $p, q$ are polynomials in $\mathbb Z[x]$. The text ...
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Can we usefully characterize those rings whose every non-zero prime ideal is maximal?

(All my rings are commutative and unital.) Here the following question is asked: Suppose $R$ is a ring and every prime ideal of $R$ is also a maximal ideal of $R$. Then what can we say about ...
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Express $[RG,RG]$ as $R-$ linear span of $[g,h]$

Let $R$ be a ring and $G$ be a group and $RG$ be the group ring. Denote by $[R,R]$, the additive subgroup generated by all lie products $[x,y]=xy-yx , \forall\ x,y\in R$. Then how is this that $[\...
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on the infinite power of the maximal ideal

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat functions, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
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Extended real numbers as algebraic structure

I need to work with real numbers, but extended to have an additional element. This element, I denote by $\odot$ and my set is: $\mathbb{R}_{\odot}=\mathbb{R}\cup\{\odot\}$. This element should behave ...
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Rank of a matrix over a principal ideal domain

I apologize if my question is stupid but I'm not very familiar with matrices over a principal ideal domain $R$ (For example, $R=\mathbb{Z}$ or $R=\mathbb{R}[X]$). I was wondering how to define the ...
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$c$ is irreducible implies that $\langle c\rangle $ is a maximal ideal, proof verification

This is my own personal proof, it seems right, but I want to make sure I don't carry incorrect logic into my future work. Let $R$ be a PID and $c\in R$ If $c$ is irreducible, then $\langle c\...
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Concerning the ring of all real valued functions of bounded variation on $[a,b]$

Let $B[a,b]$ the ring of all real valued functions of bounded variation on $[a,b]$ . What is the cardinality of $B[a,b]$ ? How does the maximal ideals of $B[a,b]$ look like ? How does the prime ideals ...
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Stupid question about a morphism between rings

Let $k$ be an algebraically closed field and let $\mathfrak m$ be a maximal ideal of $k[T_1,\ldots,T_n]$. By the weak Nullstellensatz we know that $k[T_1,\ldots,T_n]/m\cong k$, so by composition (with ...
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Factor rings $R/R$ and $R/0$

Let $R$ be a ring. I want to describe the factor rings $R/R$ and $R/0$. So $R/R = \{[r]| r+R, \forall r\in R \}$ and since $r+R=R$, we get $R/ R =\{[0]\}$. And for $R/0 = \{[r]| r+0,\forall r\in R\}$...
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Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
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What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like?

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like? And also what would the ring $\mathbb{Z}[x,y]/(x^3-x-y^2)$ look like? These are two sorts of rings I have been curious about.
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Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
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Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
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Proof review: Every maximal ideal of ring of continuous functions has the same form

Let $R$ be the ring of real-valued continuous functions on $[0,1]$. If $M$ is a maximal ideal of $R$ prove $\exists \lambda \in [0,1]$ s.t. $M = \{f(x) \in R : f(\lambda) = 0 \}$. (from Herstein ...
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Maximal ideals of R[x]/(f(x))

I have been studying for my Qualifying Exam and came across the following problem: Let $R\subseteq T$ be integral domains and suppose that $a\in T$ satisfies a monic polynomial of degree $d$ with ...
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A field F can be extended to one in which all polynomials over F have a solution

Please check to see if my methods are correct. This question is a lot to absorb all at once. Let P be the set of non-constant polynomials over a field F and let X be the set of finite subsets of P....
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Inverse limits of quotient rings

Let $A\subset B$ be an extension of discrete valuation rings and let $p$ and $P$ be the non-zero prime ideals of $A$ and $B$ respectively. So I can write $pB=P^m$ for some $m>0$. I form the ...
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Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
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Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = \...
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Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
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Uniqueness of the decomposition of an ideal

Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ I=\prod_{...
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Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
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Ring structure of a localization

let $R$ be a commutative noetherian ring and let $A$ be an $R$-algebra which is moreover a finitely generated $R$-module. Let $P$ be a prime ideal of $R$. How is the ring structure of the localization ...
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Idempotent semiring

Let $R$ be a semiring. For $a\in R$,we define $t_a(x)=x+a$ for all $x\in R$. Prove that $R$ is idempotent(with +) and $1$ has an infinite order if and only if for all $a,x,y\in R$, $t_a(x+y)=t_a(x)...
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Showing function defined on $\text{Frac}(R)$ is a ring homomorphism

Let $f : R \to S$ be a ring homomorphism where $R, S$ are integral domains. I want to show that $\varphi : \text{Frac}(R) \to \text{Frac}(S)$ defined by $r/1 \mapsto f(r)/1$ is a ring homomorphism. ...
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Inverse limit of the following system.

Let $\mathcal{O}_L = \mathbb{Z}[\sqrt{-5}]$ be the number ring of $L=\mathbb{Q}[\sqrt{-5}]$, and $\mathfrak{p} =(2, 1+\sqrt{-5})$ a prime (and maximal ideal) in $\mathcal{O}_L$. What is the inverse (...
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Dimensions of quotient rings of $K[x,y]$

I have tried to solve the following problem and would be very grateful if someone could check my answer. Let $K$ be an algebraically closed field with $\mathrm{char}(K)=0$. I wish to compute $\...
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Rings where action of automorphisms on maximal ideals is transitive

If $R$ is a commutative ring, $\alpha: R \to R$ an automorphism of $R$, and $M$ a maximal ideal of $R$, then $\alpha(M)$ is also a maximal ideal of $R$ with the same quotient field. So the group of ...
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Prove $λ$ must divide one of $x$, $y$, or $z$.

Let $λ = (3 + \sqrt{−3})/2 \in \Bbb Q[\sqrt{−3}]$. Prove that if $x^3 + y^3 = z^3$, and $x$, $y$, $z$ are quadratic integers in $\mathbb Q[\sqrt{−3}]$, then $λ$ must divide one of $x$, $y$, or $z$. ...
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Definition with Euclidean domain

Let $R$ be a Euclidean domain and let $A$ be an ideal of $R.$ Then there exists an element $a_0 \in A$ such that $A$ consists of all $a_0x$ as $x$ ranges over $R.$ I found the above theorem ...
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Characterization of epic morphisms in the category of rings.

Let $C$ denotes the category of rings (with identities). It's easy to prove that for any $R\in \text{ob }C$, any multiplicative set $S\subset R$ and any ideal $I\leqslant R$, both $R\to S^{-1}R$ and $...
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Maximal ideals of polynomial ring $\mathbb Z_n[x]$

How would I find one? say $n=p^2q^2$ for $p,q$ primes
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Question about the Chinese Remainder Theorem and the residue class ring ${\bf Z}/p\# {\bf Z}$

In a question that I asked on MO, Terence Tao observed that: The Chinese Remainder Theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of ...
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285 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
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Why is every non-zero element not a unit of this ring?

Consider the ring $\mathbb{Z}[\sqrt2]=a+b\sqrt2$ where $a,b\in\mathbb{Z}$. Now, if I am understanding the definition of units correctly, they are all the elements within the ring that have ...
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Euclidean domains and Fields

I've been wrtiting a chain of inclusions of algebraic structures as given at the end of this first paragraph on wikipedia: http://en.wikipedia.org/wiki/Euclidean_domain And I've been giving examples ...
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Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor.

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor. My answer goes like this: If ab is a zero-divisor, then there exists a nonzero ...
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58 views

On units in subrings and quotient rings of a finite ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal (or subring). Let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$. Then does $|S^*|$ divide $|R^*|$ ? Moreover, if $I$ is an ideal of ...
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Construct the rule in fields extensions

Let $f(x) = x^3+6x^2-12x+3$ Show that f(x) is irreducible over $\mathbb Q$ . Let $\theta$ be a real root of $f(x)$, which exists due to the intermediate value theorem. $\mathbb Q(\theta)$ consists of ...
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What are the ideals of $End(R)$?

Let $R$ be a ring (with unity if necessary) , then $End(R)$ i.e. the set of endomorphism of the ring $R$ (the set of all ring homomorphisms from $R$ to $R$ ) forms a ring under point-wise function ...
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Example of a maximal idea

Let $A$ be the set of bounded continuous functions from the set of real numbers to itself. Then $A$ is a ring under pointwise addition and multiplication. The set $I$ of all functions $f \in A$ ...