This tag is for questions about rings, which are a type of structure studied in abstract algebra and algebraic number theory.
5
votes
3answers
85 views
Ideal generated by a set in a commutative ring without unity
In a commutative ring with unity $1$, call it $R$, the the ideal generated by the set $S=\{a_1,...,a_n\}$ is the smallest ideal of $R$ containing $S$. It can be proven that this ideal is
$$
...
4
votes
0answers
52 views
A question on an answer on Math Overflow about Artin approximation
I have a question on an answer of this Math Overflow question.
Let $(A,I)$ be a commutative excellent normal local domain. The completion
$$
\hat A=\underset{\longleftarrow}{\operatorname{lim}} ...
2
votes
1answer
56 views
Noetherian and Artinian modules over subrings
I have a question about whether Noetherian-ness and Artinian-ness of modules are preserved under changes of the base ring. More precisely:
Let $R$ be a commutative ring and $S \subseteq R$ a ...
2
votes
1answer
37 views
The local cohomology modules are Artinian
Let $(R,m,k)$ be Noetherian local ring and $M$ a finitely generated $R$-module. Lemma 3.5.4 of Bruns-Herzog states that
the local cohomology modules $H^i_m(M)$ are Artinian
and that this ...
2
votes
1answer
45 views
ring isomorphism in the complex numbers
Let $f:\mathbb{C} \to \mathbb{C}$ be a ring isomorphism for which $f(x) = x$ for all $x\in \mathbb{R}$. Prove that $f$ is either the identity mapping ($\mathrm{id}:\mathbb{C} \to \mathbb{C}$) or f ...
2
votes
1answer
33 views
The set of zero divisors is the union of radicals of annihilators
I am trying to figure out why the statement
$$\text{the set of zero divisors }=\bigcup_{0\ne x\in R} \sqrt{\text{Ann}(x)}$$
is true. Here $R$ is a commutative ring, $\text{Ann}(x)=\{r\in R\mid rx=0\}$ ...
2
votes
2answers
28 views
Ring $\mathbb{Z}/2mnr \mathbb{Z}$ unit, identity, orders
Let $p$ be a prime number which doesn't divide $2mnr$. So $p$ is a unit in the ring $\mathbb{Z}/2mnr \mathbb{Z}$ and $q=p^k$ for a certain $k \in \mathbb{Z}$
Could you explain to me why then:
1) ...
4
votes
1answer
57 views
An ideal which is zero when squared
Suppose that $I^2=0$ implies $I=0$ for $I \triangleleft R$, $R$ a ring (not necessarily with identity).
I want to prove that assuming this condition, the condition also holds for one sided ideals ...
2
votes
3answers
58 views
Identity of Rings
How would I show that there is a ring $R$ with identity $1_R$ and a subring $S$ not containing $1_R$, but such that $S$ has its own identity $1_S$ not equal to $1_R$?
6
votes
4answers
88 views
Definition of Jacobson radical
This may be a rather silly question, but I wonder why the definition of the Jacobson radical always is
$$\{x\in R\mid 1-xy \text{ is a unit for all } y\in R\}$$
and not
$$\{x\in R\mid 1+xy \text{ is ...
2
votes
2answers
59 views
Principal Ideal Domain
Let $D$ be a principal ideal domain and let a be some fixed element of $D$. Let $(a)$ denote the ideal generated by $a$. Prove that if $a$ is irreducible and $I$ is an ideal of $D$ such that ...
-4
votes
2answers
60 views
all the maximal ideals of $I=\mathbb Z_8\oplus\mathbb Z_{30}$ [closed]
How to find all the maximal ideals of $I=\mathbb Z_8\oplus\mathbb Z_{30}$ and for each such ideal $I$ how to find the size of $R/I?$
0
votes
1answer
23 views
How to calculate the quotient and the reminder when $F=\mathbb Z_5[x],f(x)=3x+1,g(x)=x^3+2x+1?$
Division Algorithm says that, for any field $F$ and for $f(x),g(x)(\neq 0)\in F[x]$$~\exists$ unique $q(x),r(x)\in F[x]$ such that $f(x)=g(x)q(x)+r(x)$ where $\deg r(x)<\deg g(x)$ or $r(x)=0.$
I'm ...
2
votes
1answer
34 views
On regular elements and Maximal Cohen-Macaulay modules
I was reading theorem 3.3.3 in Bruns-Herzog: we have a Cohen-Macaulay local ring $(R,\mathfrak m,k)$, $C$ and $M$ are maximal Cohen-Macaulay modules. (Probably to solve my question some of these ...
3
votes
0answers
55 views
Hilbert’s zeros theorem, an application. (The algebraic variation)
Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
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
29 views
If $x_i$ generate an $A$-module $M$, why do $1 \otimes x_i$ generate the extension of scalars $B \otimes_A M$?
In the following, let "ring" be a synonym for "commutative ring with identity".
For rings $A, B$ and an $A$-module $M$, let $M_B = B \otimes_A M$ be the $B$-module obtained from $M$ by extension of ...
0
votes
2answers
32 views
Zero Divisor Rings
Let $R$ be a ring, and let $a,b \in R$ such that $ab \ne 0$. Show that $ab$ is a zero divisor if and only if $a$ is a zero divisor or $b$ is a zero divisor.
I understand that a zero divisor is $ab = ...
0
votes
0answers
17 views
normed division algebra
Can we prove that every division algebra over $R$ or $C$ is a normed division algebra?
Or is there any example of division algebra in which it is not possible to define a norm?
Definition of normed ...
4
votes
1answer
62 views
Why are quotient modules $M / \mathfrak{m}M$ over residue fields $A / \mathfrak{m}$ considered for local rings rather than general rings?
In the following, let "ring" be a synonym for "commutative ring with identity". In the book on Commutative Algebra by Atiyah and MacDonald, I read:
Let $A$ be a local ring, $\mathfrak{m}$ its ...
0
votes
2answers
97 views
How to show that $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 5)$ are non-isomorphic?
How to show that $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 5)$ are non-isomorphic as a ring?
All I could manage to show is that,
for any isomorphism $\phi:\mathbb(\sqrt 2)\to\mathbb(\sqrt 5),$ ...
1
vote
0answers
19 views
Triangular rings with direct sums
In the lam book ( a first course in non commutative rings),
He is representing the triangular ring with direct sum!!
I could not understand this part?
How can we consider the triangular rings with ...
1
vote
0answers
32 views
Frobenius from Hurwitz's theorem
Can we deduce Frobenius theorem from Hurwitz's theorem on Normed division algebra?
Frobenius theorem states that the only associative finite dimensional division algebras over the real numbers are R, ...
2
votes
5answers
47 views
Prove Principal Ideal Domain from Bezout's condition, and terminating divisibility chain
The following is a problem from Dummit & Foote.
Let $R$ be an integral domain. Prove that if the following two conditions are true, then $R$ is a principal ideal domain.
Any two non-zero ...
0
votes
0answers
32 views
Structure theorem of finite rings
Like structure theorem for finite abelian groups or modules over PID, is there any structure theorem for finite rings? Thanks.
4
votes
1answer
49 views
Sufficient condition for commutativity of finite rings
We know from Wedderburn's theorem any finite division ring is necessarily commutative.
Is there any other condition on finite rings which forces the ring to be commutative?
1
vote
2answers
43 views
A complete set of orthogonal idempotents in a commutative ring
I'm reading David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. At page 13, Chapter $0$, he says: "... if $e_1,\ldots,e_n$ is a complete set of orthogonal idempotents in a ...
3
votes
2answers
72 views
Question about the radical of the Jacobson radical.
I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
0
votes
0answers
77 views
How to show an ideal is zero-dimensional? [duplicate]
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional.
How do I go about showing this?
5
votes
1answer
49 views
For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?
I'm reading the Atiyah-MacDonald book on Commutative Algebra.
At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is:
$G$ = finite group, ...
1
vote
0answers
56 views
Another problem with a proof in “Topics in Algebra” by Herstein- every element in R can be written as a finite product of prime numbers.
On pg. 146 of the second edition, it says let $a=bc$, where $a,b,c\in R$. $R$ is a Euclidean ring.
If $d:R\to Z$, then if $b$ is a unit, $d(a)=d(c)$ [$d(a)=d(b)$ if $c$ is a unit]. If neither $b$ nor ...
0
votes
1answer
30 views
Problem with proof in “Topics in Algebra” by Herstein.
On pg. 148 of the second edition, the proof for the unique factorization theorem in Euclidean rings is given.
Let $\pi_{1}.\pi_{2}.\pi_{3}\dots = \pi_{1}'.\pi_{2}'.\pi_{3}'\dots\dots$, where all the ...
0
votes
1answer
30 views
Showing that the ring of $n\times n$ matrices has exactly two 2 sided ideals, even though it is not a division ring
Show the ring $A=\mathrm{Mat}(F)$ has exactly two 2-sided ideals, even though it is not a division ring.
$F$ is any field, and $\mathrm{Mat}(F)$ is all $n\times n$ matrices with elements of $F$ ...
1
vote
2answers
70 views
Find all maximal ideals of $\mathbb{Z}_{540}$
Find all maximal ideals of $\mathbb{Z}_{540}$
By using the following statement,
'$f:R \rightarrow S$ be a surjective ring homomorphism and let $K=ker(f)$.
Observe that there is a one-to-one ...
1
vote
2answers
27 views
Examples of $F$-algebra which has no proper two-sided ideals but has one-sided ideal(s)
Let $F$ be a field and $A$ an $F$-algebra. (And assume that $A$ is finite dimension over $F$ if necessary.) A textbook says that $A$ is simple if it has no proper two-sided ideals.
To understand this ...
8
votes
1answer
80 views
Commutativity characterization?
Let $R$ be a ring (not necessarily unital) and for any $x\in R$ there is an integer $n \geq 2$ s.t. $x=x^2+\cdots+x^n.$
Does it imply that $R$ is commutative?
3
votes
4answers
85 views
How to find all the maximal ideals of $\mathbb Z_n?$
How to find all the maximal ideals of $\mathbb Z_n?$
I think $(0)$ is the only maximal ideal of $\mathbb Z_n$ for if $a$ is a non-unit in a maximal ideal of $\mathbb Z_n$ then ...
1
vote
1answer
60 views
Krull Dimension
I'm studying Krull dimension and I'm confused about the definition of $\text{ht}(P)$, which is as I understand is the following: let $$P_0\subset P_1\subset\dots\subset P_n=P$$ be a chain of prime ...
2
votes
2answers
49 views
Are there analogues of eigenvalues/eigenvectors for a ring homomorphism/endomorphism?
My question is very simple. To put it in a context, a linear transformation is nothing but a homomorphism from a vector space to another. I usually visualize the action of a linear transformation by ...
0
votes
0answers
29 views
What is 0 mapped to in a Euclidean domain?
Let us suppose we have a Euclidean domain A, in which we have $a=q*d+r$. We know that there is a function $f:A\to Z$ such that for every $a\in A/0$, we have $f(a)>f(0)$. Also, $f(r)<f(d)$.
Is ...
0
votes
0answers
12 views
Finite generation of a subalgebra of R[x]
Let $k$ be a field, $R$ a noetherian $k$-domain, $x$ a central variable over $R$ and $A$ a (commutative) $k$-subalgebra of $R[x]$. So $R[x]$ is an Ore domain. Now, suppose that there exists a finitely ...
1
vote
1answer
51 views
Field extension of $\mathbb Q$ of degree 2
Assume $K:\mathbb Q$ is a field extension and $[K:\mathbb Q] = 2$. Show that there is a unique squarefree $d \in \mathbb Z$ such that $K = \mathbb Q(\sqrt d)$.
I know that $K$ is generated by say ...
6
votes
0answers
70 views
An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
1
vote
1answer
38 views
Do ideals partition a ring?
Say we have two principal ideals- $(a)$ and $(b)$. Is $r_{1}*a=r_{2}*b$ possible for $r_{1},r_{2}\in R$, with $(a) \neq (b)$?
I don't see a problem with this as long as the multiplicative inverses of ...
1
vote
0answers
43 views
What is $\mathbb{C}[xy]/\langle x\rangle \subseteq \mathbb{C}[x,y]/\langle x \rangle$?
Consider the ring $\mathbb{C}[x,y]$,
and consider
$$R=\dfrac{\mathbb{C}[xy]}{\langle x\rangle } \subseteq \dfrac{\mathbb{C}[x,y]}{\langle x\rangle }\cong \mathbb{C}[y].$$
Is $R\cong ...
3
votes
1answer
43 views
Correspondence between submodules and quotient modules
What is the (natural) bijection between the set of all sub modules upto isomorphism and set of all isomorphic quotient modules upto isomorphism of a finitely generated torsion module over a PID. Is ...
2
votes
1answer
66 views
Prove that $D[x]$ is an integral domain if $D$ is one.
Prove if $D$ is an integral domain and $f,g\in D[X]$ are nonzero, then $fg$ does not equal $0$ and $\deg[f(x)g(x)]=\deg f(x) + \deg g(x)$.
I do not know much about this since I just learned about it. ...
2
votes
0answers
36 views
Modules with maximal submodules and projective dimension
If $R$ is a left noetherian ring, then every finitely generated left $R$-module $M$ is noetherian, and hence every proper submodule of $M$ is contained in some maximal submodule of $M$.
Is it ...
3
votes
3answers
129 views
Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.
I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
2
votes
1answer
31 views
Morita contexts and Noetherianity/affineness
Let $(R\,,\, S\,,\, _RM_S\,,\, _SN_R\,,\, f\,,\, g)$ be a Morita context with $NM=S$ and $R$ right Noetherian. Show that $S$ is right Noetherian as well. If we further assume $R$ is an affine ...
