This tag is for questions about rings, which are a type of structure studied in abstract algebra and algebraic number theory.
3
votes
2answers
61 views
Non-isomorphic rings of given cardinality that are non-commutative
I need to find an example of two non-isomorphic rings of cardinality 16 that are non-commutative. What is the best approach to such problem?
3
votes
2answers
210 views
A formula for the minimum number of generators of a module
Let $R$ be a commutative ring with only finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_r$. Let $M$ be a finitely generated $R$-module. Then
$$\mu_R(M)=\max\{\dim_{R/\mathfrak ...
2
votes
2answers
92 views
homomorphisms and product rings
The problem is this: Let $f:\mathbb{R}[x]\rightarrow \mathbb{C}\times \mathbb{C}$ be the homomorphism defined by $f(x)=(1,i)$ and $f(r)=(r,r)$, for $r\in \mathbb{R}.$ Determine the kernel and the ...
0
votes
2answers
25 views
In $Z_5 [x]$, the monic GCD of the polynomials $(x+[4])(x+[3])$ and $([3]x + [2])(x + [3])$ is $(x + [3])$
True or False
In $Z_5 [x]$, the monic GCD of the polynomials $(x+[4])(x+[3])$ and $([3]x + [2])(x + [3])$ is $(x + [3])$.
my solution :
$([3]x+[2])$ is $[3](x+[4])$ therefore gcd is ...
-1
votes
2answers
83 views
An example of a divisible ideal
Let $R$ be a commutative ring (other than a field) with identity. I am looking for an example of a divisible ideal of $R$.
1
vote
1answer
48 views
Localisation and extension of rings
Is $\mathbb{Z}_{(3)}[i,\sqrt{2}]=(\mathbb{Z}[i,\sqrt{2}])_{(3)}$ (where by subscript $(3)$ we mean localisation at the ideal generated by $3$)?
Do both of these rings contain elements like
$$
...
1
vote
1answer
54 views
Are these prime ideals?
Let $R=\mathbb Z[\sqrt{-5}]$. I want to show $P=3\,R+(1+\sqrt{-5})\,R$ and $Q= 3\,R+(1-\sqrt{-5})\,R$ are prime ideals of $R$.
1
vote
1answer
38 views
Prime and semiprime ideals of $A=T_3(D)$, the ring of $3\times 3$ upper triangular matrices over $D$
Let $D$ be a division ring. Could anyone tell me which are the prime and semiprime ideals of $A=T_{3}(D)$, where $A=T_{3}(D)$ the ring of $3\times 3$ upper triangular matrices with coefficients in ...
1
vote
1answer
67 views
Artinian ring with zero finitistic dimension
Let $R$ be a left artinian ring with identity.
Suppose $R$ contains copies of all its simple right $R$-modules.
Is it true that every left $R$-module of finite projective dimension is projective (so ...
1
vote
1answer
55 views
Finding ring endomorphisms.
I need to find $\varphi \in \operatorname{End}(\mathbb{R}[x])$ such that there's a function $\psi \in \operatorname{End}(\mathbb{R}[x]), \psi \neq 0$ such that $\psi \circ \varphi = 0$ but there's no ...
1
vote
1answer
68 views
Why is multiplication well defined in this ring with the Ore condition?
I'm reading Linear Equations in Non-Commutative Fields by Oystein Ore in the Annals of Mathematics. The papers is available here or here if one has access to jstor.
Ore is working in a ...
0
votes
1answer
58 views
Matrix ring over a field and its ideals
Let $M_n(R)$ be the matrix ring over a commutative ring $R$ and let $I$ be an ideal of $R$.
1) Show that if $R=F$ is a field then the only nonzero ideal of $M_n(F)$ is $M_n(F)$ itself.
2) Let ...
0
votes
1answer
47 views
Ring Isomorphism Proof
Let $p$ be a prime with $p \equiv 1 (\mod 4 )$.
I am trying to show that $\mathbb{Z}[X]/(X^2 + 1, p) \cong \mathbb{Z}_p \times \mathbb{Z}_p$ is a ring isomorphism.
I am not really sure how to ...
0
votes
1answer
33 views
Unique non trivial ideal in CFM(k)
Given CFM(k), the ring of infinite matrices with finite columns, I've already proved that the subset of matrices with a finite number of rows no null is an ideal.
How can I prove the following ...
0
votes
1answer
52 views
Smallest Subring
Suppose that $S$ and $T$ are subrings of a ring $R$. Show that their ring-theoretic product $ST$ is a subring of $R$ that contains $S \cup T$, and is the smallest such subring.
I understand that $ST$ ...
0
votes
1answer
94 views
Maximal ideals of some localization of a commutative ring
If $R$ is commutative ring, $P_1, P_2, ..., P_n$ prime ideals of $R$ with the property $\forall 1≤i≤n : P_i \not\subset \bigcup _{j \not = i} P_j$ and $S:=R\setminus(P_1 \bigcup ... \bigcup P_n)$, ...
0
votes
1answer
161 views
homomorphism polynomial ring
1/ If $d$ is not a square in $\mathbb{Q}$, show that $\mathbb{Q}[\sqrt{d}]\approxeq\mathbb{Q}[X]/<X^2-d>$ where $(X^2 - d)$ is the principal ideal of $\mathbb{Q}[X]$ generated by $X^2 - d$.
2/ ...
0
votes
1answer
30 views
Expressing $I/J$ in terms of quotients of the larger ring
Let $R$ be a Noetherian ring. Let $I\subseteq J$ be nonzero left ideals of $R$. Can the factor ring $I/J$ be expressed in terms of sums, quotients or submodules of rings of the form $R/K$, where $K$ ...
-1
votes
1answer
41 views
Splitting field questions.
$K$ is a field and $f \in K[x]$ with splitting field $L$. Show that $[L:K] \le n!$, where $n$ is the degree of $f$.
$f \in \mathbb{Q}[x]$ is a cubic polynomial and $K$ is its splitting field. What ...
-1
votes
1answer
100 views
Noetherian prime ideal is $R$-module and $R$ is Noetherian ring
If $R$ is a commutative ring, and for every $P$ a prime ideal of $R$, $P$ is a Noetherian $R$-module, show that $R$ is Noetherian.
-2
votes
1answer
120 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 ...
-2
votes
1answer
60 views
true or false questions on prime and maximal ideals
which of the following are true and which are false:-
$1.$ Every prime ideal of every commutative ring with unity is a maximal ideal.
$2.$ The prime subfield of $\mathbb{C}$ is $\mathbb{C}$
$3.$ ...
-4
votes
1answer
133 views
$\mathbb{Z}[\sqrt{d_1}]$ and $\mathbb{Z}[\sqrt{d_2}]$ are not isomorphic
If $d_1, d_2$, and $d_1/d_2$ are not squares in $\mathbb{Z}\backslash\left\{ 0\right\} $, show that $\mathbb{Z}[\sqrt{d_1}]$ and $\mathbb{Z}[\sqrt{d_2}]$ are not isomorphic
