This tag is for questions about rings, which are a type of structure studied in abstract algebra and algebraic number theory.
-1
votes
0answers
18 views
find a special element in group algebra
Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual ...
6
votes
2answers
48 views
Under what conditions does a ring R have the property that every zero divisor is a nilpotent element?
Under what conditions does a ring $R$ have the property that every zero divisor is a nilpotent element ?
If we have a ring $R$, we know that every nilpotent element is either zero or a zero divisor.
...
3
votes
1answer
30 views
About generators of a finitely generated ideal
Let $R$ be a ring with $1$. Let $S$ be a subset of $R$, with infinitely many elements. Let $\mathfrak{i}$ be the ideal of $R$ generated by $S$. Suppose $\mathfrak{i}$ finitely generated:
...
-2
votes
1answer
58 views
Question about polynomial rings.
If $F[x]$ is a polynomial ring, and $f(x), g(x), h(x)$ and $r(x)$ are four polynomials in it, then is it always true that $f(x)=h(x)g(x)+r(x)$ where $deg(r(x))<deg(g(x))$, or is this true only when ...
5
votes
2answers
97 views
Seeing that $\Bbb F_2[x]/(x^2+x+1)$ is a field
I have some basic question with polynomials appreciate if someone could explain me this.
Following is additional and multiplication tables and it is say that this is a field. Have no idea why say ...
6
votes
3answers
66 views
Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies
Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime.
Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
0
votes
1answer
16 views
Tensor product of two cyclic modules
Given a commutative ring $A$ (with identity) and two cyclic $A$-modules, $M$ and $N$ with generators $x$ and $y$, respectively.
How do you show that $\mathrm{Ann}(x\otimes_A y) = \mathrm{Ann}(x) + ...
3
votes
1answer
36 views
Jacobson radical of non-unital rings
This question come from a colleague who does operator theory, and hence works with non-unital rings, which I know little about. It's well-known that any non-unital ring $R$ can be embedded in a unital ...
0
votes
3answers
30 views
Prime ideal and nilpotent question
If $p \subset R$ is a prime ideal, prove that for every nilpotent $r \in R$, it follows that $r \in p$.
The only hint that my tutor gave me was to use induction. Can someone explain what he means by ...
4
votes
0answers
45 views
Deciding whether or not a class of modules is “big enough”
For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
-3
votes
1answer
65 views
$\mathbb{Z}[\sqrt{-23}]$: A uniquely written set?
I suspect that
$\mathbb{Z}[\sqrt{-23}] \implies \forall~z=\sqrt{23b+a}~e^{i\arctan{\frac{23b}{a}}},~\text{where $z$ is uniquely written}~\forall~z\in \mathbb{Z}[\sqrt{-23}]$
2
votes
3answers
58 views
$\mathbb{Z}/m\mathbb{Z}$ is free when considered as a module over itself, but not free over $\mathbb{Z}$.
I am struggling to explain the following statement to myself in a convincing way. $\mathbb{Z}/m\mathbb{Z}$ is free when considered as a module over itself, but is not free when considered as a ...
1
vote
0answers
30 views
Equality of two $k$-algebras
Let $f\in k[X_1,\ldots, X_n]$ and $1-fX_{n+1}\in k[X_1,\ldots, X_{n+1}]$. Moreover $X\subseteq k^n$ is a subset and
$$I(X)=\{g\in k[X_1,\ldots, X_n]\,:\, g(x)=0\,\forall x\in X \}$$
is the ideal of ...
3
votes
1answer
38 views
$S^{-1}B$ and $T^{-1}B$ isomorphic for $T=f(S)$
Let $f:A\to B$ be a homomorphism of rings, $S$ be a multiplicatively closed subset of $A$ and $T=f(S)$. Then $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules.
First we define the ...
3
votes
2answers
25 views
Every ideal of the localization is an extended ideal
Let $R$ be a ring, commutative with $1$. Let $S$ be a multiplicatively closed subset of $R$, with $0\notin S,1\in R$. Let $R_S$ be the localization of $R$ at $S$. For every ideal ...
