Questions tagged [ring-theory]
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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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Quotient of graded ring is graded - confusion about the formalisms
A ring $R$ is graded if it has a direct sum decomposition $R=\bigoplus_{i\in\mathbb{Z}}R_i$ where the $R_i$ are abelian groups and $R_iR_j\subseteq R_{i+j}$. An ideal $I\subseteq R$ is graded if $I=\... abstract-algebra ring-theory graded-rings- 13
Existence of maximal ideal in a commutative ring
Let $A$ be a commutative ring, $I \subsetneq A$ a proper ideal of $A$ and $a \in A$ such that $a^k \neq 0$ for all integer $k > 0$. Then there exists an ideal $J$ of $A$ that is maximal satisfying $... abstract-algebra ring-theory maximal-and-prime-ideals nilpotence- 1,030
To prove $ \Bbb{Z}_p$ is isomorphic to $\lim_{←}\Bbb{Z}_p/p^n \Bbb{Z}_p$ as a ring
I want to prove $ \Bbb{Z}_p$ is isomorphic to $\lim_{←}\Bbb{Z}_p/p^n \Bbb{Z}_p$ as a ring. Here, $\Bbb{Z}_p$ is completion (as metric space) of $\Bbb{Z}$ with p adic metric. My try: Let define natural ... abstract-algebra ring-theory p-adic-number-theory limits-colimits- 29
The Making of a Ring — How can we choose 1?
I am an undergraduate studying Algebra: Chapter 0 by Paolo Aluffi in my free time (Note that because of this whenever I refer to 'rings' I am speaking of 'rings with identity'). While learning about ... abstract-algebra group-theory ring-theory- 237
Ring $\mathbb F_q[x]/\langle f^n \rangle $ where f is irreducible
We know that the ring $\mathbb F_q[x]/\langle f \rangle $ is a Galois ring when $f$ is an irreducible polynomial over $\mathbb F_q$. Is there any idea to describe the ring $\mathbb F_q[x]/\langle f^... ring-theory- 1
Prove that the ideal $\langle x^2+1\rangle$ is prime in $\mathbb{Z}[x]$.
Prove that the ideal $\langle x^2+1\rangle$ is prime in $\mathbb{Z}[x]$. $\textbf{My attempt:}$ Let $f(x)g(x) \in \langle x^2+1\rangle$, then $(x^2+1) \mid f(x)g(x)$. I want to prove that wither $f(x)... abstract-algebra ring-theory ideals maximal-and-prime-ideals- 281
What are the properties of this new characteristic of mathematical objects?
I will call it "hypermodulus". In simple words, hypermodulus is the exponent of the scalar part of the finite part of the logarithm of the object: $H(A)=\exp (\operatorname{scal} \... ring-theory regularization hypercomplex-numbers divergent-integrals umbral-calculus- 7,742
Why we cannot divide one zero divisor by another one? Or can we?
For instance, in split-complex numbers we definitely have $2\cdot(j/2+1/2)=j+1$, which is absolutely valid. Can we then say that $\frac{j+1}{j/2+1/2}=2$? If not, why we cannot define it this way? abstract-algebra ring-theory hypercomplex-numbers- 7,742
Find all power series $f$ such that $\mathbb{C}[[x, y]]/(f(x, y))\cong \mathbb{C}[[x, y]]/(xy)$
Find all power series $f$ such that $\mathbb{C}[[x, y]]/(f(x, y)) \cong \mathbb{C}[[x, y]]/(xy)$. The isomorphism $\phi$ should be identical on $\mathbb{C}$, i.e., $\forall c \in \mathbb{C} \subseteq ... abstract-algebra ring-theory contest-math- 81
Prime ideal generated with polynomial of degree 0?
Let $K=\{p\in \mathbb{Q}_n[x]\ |\ p(0)\in \mathbb{Z}\}$ be a commutative ring. I have to check if an ideal $I=\{p\in K\ |\ p(0)=0\}$ of $K$ is prime or not. I would suppose that $I$ is a prime ideal. ... abstract-algebra ring-theory ideals maximal-and-prime-ideals- 65
Basis and dimension of Ring as a vector space over finite field
I tried to solve following exercise: My knowledge in algebra is a bit rusty, so I would appreciate it if you could help me remembering how to solve c). I realized that $|R| = 2^{128}$ but I don't ... abstract-algebra ring-theory vector-spaces finite-fields- 29
Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and for every $[a]_{n}$ we get $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ .
Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and every $[a] \in \mathbb{Z}_{(m,n)}$ can be regarded as an element $[a]_{n}$ such that $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ . ... abstract-algebra group-theory ring-theory modular-arithmetic modules- 85
Prove that any ring $R$ where there exists a prime $k$ such that $kx=0$ and $x^k=x$ for all $x\in R$ is commutative [duplicate]
Prove that any ring $R$ where there exists a prime $k$ such that $kx=0$ and $x^k=x$ for all $x\in R$ is commutative I can only prove while $k=2$ or $3$: If $a\in R$ satisfied $a^2=a$, then $$\begin{... abstract-algebra ring-theory- 39
Understanding Weibel's proof of "for local ring $R$, every element $u\in R$ invertible in $R/\mathfrak{m}$ is a unit of $R$"
In Weibel's K-book, there is a lemma saying that for a local ring $R$, with maximal ideal $\mathfrak{m}$, every finitely generated projective $R$-module is free. The proof uses a simple statement that ... abstract-algebra ring-theory local-rings- 13
Ring and quotient ring
Let R be a ring and I is an ideal of R. Can quotient ring R/I has an identity but R does not? I'm having trouble with the example. Thank you so much. ring-theory- 1
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