Questions tagged [artinian]

For questions on Artinian rings, Artinian modules and related notions.

Filter by
Sorted by
Tagged with
1
vote
0answers
15 views

Any finite Abelian group is Artinian module over any ring $R$?

Finite Abelian groups are Artinian as modules over $\mathbb{Z}$ It is known that if a module $M$ has finitely many submodules has to be Artinian. So above statement can be implied by this right! Can ...
0
votes
1answer
54 views

Dimension of localizations as complex vector spaces.

Consider the ring $A=\mathbb{C}[x,y]/\left\langle x^2-y^3, 4y^2+x^4+5x^2y\right\rangle$. I proved that $A$ is a finite dimensional vector space over $\mathbb{C}$. More precisely $$A=\langle \overline{...
0
votes
1answer
71 views

If $\mathfrak{m}_1\mathfrak{m}_2\cdots \mathfrak{m}_n=0$ then $A$ is an Artinian ring.

I want to prove the following: Let $A$ be a Noetherian commutative ring with unity, and suppose that zero ideal is a product $\mathfrak{m}_1\mathfrak{m}_2\ldots \mathfrak{m}_n$ of maximal ideals in $A$...
0
votes
1answer
38 views

Grothendieck group of artinian abelian category

Let $\mathscr{C}$ be an artinian, abelian category and $K(\mathscr{C})$ its Grothendieck group. If $[A]=0$ in $K(\mathscr{C})$ for an object $A$, can we then conclude $A=0$? This is true if $\mathscr{...
0
votes
2answers
53 views

Quotient of Noetherian local ring with $\mathfrak{m}$-primary ideal

Let $A$ be noetherian local with maximal ideal $\mathfrak{m}$ and let $\mathfrak{q}$ be an $\mathfrak{m}$-primary ideal. Then why is $A/\mathfrak{q}$ an artin ring? $A/\mathfrak{q}$ is noetherian so ...
1
vote
0answers
51 views

Finding maximal ideals in an artinian ring.

Consider the ring $R:=\mathbb{C}[x,y]/\langle x^2-y^3, 4y^2-5x^2y+x^4\rangle$. I want to do the following: Show that $R$ is artinian and calculate the maximal ideals $\mathfrak{m}$ of $R$. How many ...
2
votes
0answers
28 views

Hypothesis - am I doing this right? (Local, principal ideal ring)

Let $R$ a commutative local principal ideal ring with 1 that is not Artinian. So it's Krull dimension is non zero. Let $P\subsetneq M$ a prime ideal and M the maximal ideal of R, since P and M are ...
1
vote
1answer
44 views

Can R[[X]] be artin ring?

Let R be ring. Is R[[X]] can be artin ring? In my image, both artin ring and formal power series ring are similar to field(but both are not field).
0
votes
1answer
127 views

Proof Exercise in Steps in commutative algebra [closed]

I am trying to solve this exercise from Sharp's book Steps in Commutative Algebra: Let $M$ be a finitely generated Artinian module over the commutative ring $R$. Show that $R/\mathrm{Ann}(M)$ is an ...
2
votes
1answer
46 views

Chain of equivalences in 12.B of Matsumura's "Commutative Algebra"

I'm reading Matsumura's "Commutative Algebra", specifically the chapter on Dimension. However I am having some trouble in the section 12.B, where it is presented the chain of equivalences ...
0
votes
0answers
33 views

About artinian rings and homomorphisms

Let $R$ be a left artinian ring, $\mathfrak{r}:=J(R)$. and e,f idempotent in $R$. Proof that the morphism of abelian groups $\varphi:eRf\rightarrow Hom_R(Re,Rf)$, $\varphi(erf)(r'e):=r'erf$ is an ...
0
votes
1answer
43 views

Grauert type theorem for artinian bases

We have met the following Grauert's theorem, say in Hartshorne's book. Let $\pi: \mathcal X\to B$ be a propre flat morphism with $B$ reduced. Let $\mathcal F$ be a locally free coherent sheaf over $\...
1
vote
1answer
24 views

$R$ such that $R/J$ is artinian ($J$ is the Jacobson radical) and there exists an idempotent cannot be lifted modulo $J$?

I know a few examples of rings and ideals such that there exists an idempotent cannot be lifted modulo the ideal (for instance, $\mathbb Z$ and $n\mathbb Z$). My question is: is there a ring $R$ such ...
6
votes
1answer
142 views

Artinian, but not Noetherian module: a wonderful example.

I found this wonderful example here. I did not understand some details, could you help me understand? The questions will be asked within the example. Let $p\in\mathbb{N}$ be a prime number. Consider ...
0
votes
1answer
36 views

A question about a Noetherian ring starting from a subring

Proposition. Let $A$ be a subring of $B$, $A$ be a Noetherian, and $B$ finitely generated as an $A-$ module. Then $B$ is a Noetherian ring. Proof. Since $B$ is finitely generated as an $A-$ module, ...
2
votes
1answer
228 views

Finite generation of a nilpotent ideal

Let $R$ be a ring with a unique minimal prime ideal $\mathfrak{p}$. Let us assume that every zero divisor in $R$ is nilpotent, so that $R \to R_{\mathfrak{p}}$ is injective. If $R_{\mathfrak{p}}$ is ...
0
votes
1answer
34 views

Question about notation in proof of Proposition 2.C of Matsumura's "Commutative Algebra"

I have a question about a notation in Matsumura's "Commutative Algebra". I post below a screenshot of the page in the book in which I encountered it, underlining some instances of the ...
2
votes
2answers
140 views

Is the ring $\mathbb{C}[x,y]/(x^2,y^3)$ Artinian? If not, what is its Krull dimension?

So, I know that the ring $\mathbb{C}[x,y]/(x^2,y^3)$ is Noetherian, since the ring $\mathbb{C}[x,y]$ is Noetherian. In order to prove that the ring is not Artinian, I've tried finding a prime ideal ...
1
vote
1answer
40 views

When Ideal transform functor is Artinian?

Let $R$ be a Noetherian ring and $I$ an ideal. For a finitely generated module $M$ over $R$ we define $$D_I(M)=\varinjlim_\limits{n\geq1}\operatorname{Hom}_R(I^n,M).$$ A variety of nice results about $...
0
votes
1answer
48 views

Question about simple module over semisimple Artinian ring

Let $R$ be a semisimple Artinian ring and $M$ be a simple left $R$-module. I need to show that there exists some left ideal $I$ of $R$ such that $M \oplus I \cong R$. I am not really sure how to do ...
0
votes
1answer
40 views

Zero dimensional quotient of a local Noetherian ring

Let $(R,m)$ be a local Noetherian integral domain. Let $I$ be an ideal in $R$, so $I \subseteq m$. Then $R/I$ is also a local Noetherian ring, see (and the image of any surjective ring homomorphism of ...
1
vote
1answer
109 views

Show that $A\simeq k[x_1,...,x_d]/I$

Let $k$ be a field and $A$ be an Artin local $k$-algebra such that $k\simeq A/M$. Then one fact is that $M/M^2$ is a finite dimensional $k$-vector space. I've saw that if $A =k[x]/(x^2)$ then $\dim_{...
0
votes
1answer
30 views

$A/\alpha$ is Noetherian as an $A$-module but how to get it is Noetherian as an $A/\alpha$-module?

I meet a problem in commutative algebra of Atiyah. Pprposition $6.6.$ Let $A$ be Noetherian (resp. Artinian), $\alpha$ an ideal of $A$. Then $A/\alpha$ is a Noetherian (resp. Artinian) ring. Proof. ...
0
votes
2answers
43 views

Show that, given $f:M→N$, if M is an artinian module so is N

What it says on the tin. Note that $f:M→N$ is a surjective $R-module$ homomorphism. This is what I have: Let ${I_n}$ denote a chain of submodules in $M$ such that ${I_0 \supseteq I_1 \supseteq ...}$. ...
0
votes
1answer
17 views

Show that if $P_n$ is a descending chain of submodules in $N$, then so is $f^{-1} (P_n)$ in $M$

Given $f:M→N$ surjective, show that if $P_n$ is a descending chain of submodules in $N$, then $f^{-1} (P_n)$ is a descending chain of submodules in $M$. I'm a bit lost to be completely honest. The ...
0
votes
0answers
58 views

On the generators of the injective hull of the residue field of $\mathbb C[x,y]/(x^2, xy,y^2)$

Consider the Artinian local ring $R:=\mathbb C[x,y]/(x^2, xy, y^2)$ with residue field $\mathbb C$. Let $E:=E_R(\mathbb C)$ be the injective hull (as an $R$-module) of the residue field (also known as ...
0
votes
1answer
39 views

Is there exist a hollow module which is not Artinian

A module is said to be hollow if, its every proper submodule is small, that is, for any two proper submodule $N,K$ of $M$, $N+K\neq M$. A module is said to be Artinian if it satisfies descending chain ...
3
votes
0answers
57 views

Do Artin rings contain many commutative Artin subrings?

Suppose $R$ is a Artin ring and $x\in R$. Is there necessarily a commutative subring of $R$ which is also Artin and contains $x$? What if I want it to contain finitely many commuting elements $x_1,\...
-1
votes
1answer
46 views

Noetherian and Artinian quotient

Let $I$ be a left ideal of ring $R$. My question is can we lift both Noetherian and Artinian property of $R/I$ as an $R$-module to $R$ as an $R$-module? Precisely, can we say that $R$ is a left ...
1
vote
1answer
47 views

Is $(\Bbb Q[x])_x/(\Bbb Q[x])$ an Artinian $(\Bbb Q[x])$-module?

Consider the ring $R = \Bbb Q[x]$ and the multiplicatively closed set $S = \{1, x, x^2, \ldots\}$. Let $R_x = S^{-1}R$. Consider the module $M = R_x/R$. Determine whether $R_x$ and $M$ are Noetherian/...
4
votes
0answers
50 views

Cohen structure theorem for Artinian local ring

I have an Artinian local ring $(R,\mathfrak{m})$ and I know that such a ring is complete in its $\mathfrak{m}$-adic topology. An Artinian ring also has the property that regular elements (non zero ...
2
votes
1answer
103 views

Commutative Artinian ring with infinitely many ideals.

Intro: An immediate example of Artinian ring is any ring with finitely many ideals. Recently, when I was thinking about Artinian rings, I realized that this is actually the only example I really know. ...
2
votes
1answer
37 views

A non-cyclic Artinian module has at least two distinct minimal (simple) submodule.

My Question: A non-cyclic Artinian module has at least two distinct minimal (simple) submodule. My attempt: Let $M$ be a non-cyclic Artinian $R$-module. Then there exists two nonzero element $x,y\in M$...
0
votes
0answers
83 views

An artinian ring is Noetherian

I am trying to understand the proof for Artinian $\implies$ Noetherian given in chapter 16 of "Abstract Algebra (3rd edition)" by Dummit and Foote. The proof given in the textbook goes as ...
0
votes
0answers
45 views

Find composition series of Αrtinian ring, that is finite dimensional vector space [duplicate]

I am trying to solve the following problem, so some help would be appreciated. Prove that the ring $R=k[t^2,t^3]/(t^4)$ is Artinian and find a composition series for the ring. Where $k$ is a field and ...
0
votes
1answer
114 views

Prove that the ring is Artinian and find a composition series of ideals

I came across the following problem Let $k$ be a field and $R=k[t^2,t^3]/(t^4)$. The ring $k[t^2,t^3]$ is defined as the set $\{f(t^2,t^3)|f \in k[x,y]\}$ with the obvious ring structure. prove that ...
0
votes
1answer
76 views

Question on injective envelopes

Let $A$ be an Artin algebra and let $\text{mod}(A)$ denote the category of finitely generated left $A$-modules. Let $S$ be a simple module in $\text{mod}(A)$ and let $\iota_S: S \rightarrow I(S)$ be ...
1
vote
1answer
55 views

An Exercise on projective covers and injective envelopes of simple modules over an Artin algebra

Let $A$ be an artin algebra and let $\text{mod}(A)$ denote the category of finitely generated left $A$-modules. Let $S$ be a simple module in $\text{mod}(A)$. Let $\iota_S: S \rightarrow I(S)$ and $\...
5
votes
1answer
81 views

Does a finitely generated faithful module over an Artinian ring contain a regular element?

In the text Nicholson -- Introduction to Abstract Algebra, 4th Ed (2012) the claim of exercise $8(b)$ of exercise set $11.1$ is: If $R$ is a left artinian ring with $1\ne 0$, and $M$ is a finitely ...
1
vote
1answer
93 views

The ring $\Bbb Q[x,y,z]/(x^a,y^b,z^c)$ is noetherian

How to show the ring $\Bbb Q[x,y,z]/(x^a,y^b,z^c)$ is noetherian? I know this ring is artin,and we can conclude the ring is artin because every ration ring is Noether.But I want to show this by ...
2
votes
1answer
49 views

Understanding the Proof of Maschkes's Theorem

I am currently reading "A Course in Ring Theory" by Passman, and I know other questions about this proof have been asked here before, but I really want to understand the proof in the book. ...
1
vote
1answer
62 views

Commutative local Artinian ring without a section in characteristic $0$

Let $R$ be a commutative local Artinian ring, with unique maximal ideal $\mathfrak m$ and residue field $k = R/\mathfrak m$. If $R$ is a finitely generated algebra over a perfect field $F$, then the ...
1
vote
1answer
85 views

How to show $\operatorname{ann}(M) = \operatorname{ann}(X)$.

If $R$ is left Artinian, $X$ a subset of a left $R$-module $M$, define $\operatorname{ann}(X) = \{a \in R: ax = 0, \forall x\in X \}.$ (a) Show $\operatorname{ann}(M) = \operatorname{ann}(X)$ for some ...
1
vote
1answer
19 views

for an endomorphism $f$ over an Artinian module

For an endomorphism $f$ over an Artinian Module $M$ over a ring $R$ i have to show that $M = f^n(M) + ker(f^n)$, we get quickly from one of the isomorphism theorems that $f^n(m) \simeq M/ker(f^n)$ can ...
1
vote
1answer
77 views

Is the injective envelope of the residue field reflexive?

Let $(R , \frak m)$ be an Artinian local ring and let $E$ denote the injective envelope of $R / \frak m$. If ${\rm Ext}_R^i(E,R)=0$ for $i>0$, then must $E$ be reflexive, that is, $E \simeq {\rm ...
1
vote
0answers
83 views

Colon property in artinian local Gorenstein

I have this problem about a property of Gorenstein artinian rings: Let $(A,m)$ be an artinian local Gorenstein graded ring such that $A_s\neq 0$ and $A_{s+1}=0$ where $A_i$ is the degree $i$ part of $...
1
vote
1answer
119 views

Is an artinian subring of a local artinian ring local?

If $R$ is a local left artinian ring, and $A$ is a left artinian subring, is $A$ a local ring? I can show that $rad A = A \cap rad R$ since $rad A$ and $rab R$ nilpotent ideals ($R$ and $A$ are ...
1
vote
1answer
30 views

$\Lambda/\mathfrak{r}\cong \operatorname{soc}(\Lambda)$ as a criterion for self-injectivity

I've been working through the exercises of Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and have gotten stuck on their Exercise 4.12, which asks: Let $\Lambda$ be an Artin ...
3
votes
1answer
128 views

Short exact sequence of modules over Artinian local ring

Let $(R,\mathfrak m, k)$ be an Artinian local ring. So for every non-zero finitely generated $R$-module $M$, we have $\mathfrak m\in \mathrm{Ass}(M)$, hence we have an exact sequence $0\to k\to M$, so ...
5
votes
1answer
131 views

Prove that for a commutative Noetherian ring $A$ with $\mathrm{Spec}(A)$ finite and discrete, $\ker(f_r)=\{0\}$ implies $f_r$ is surjective.

Let $A$ be a commutative Noetherian ring with unity with $\mathrm{Spec}(A)$ finite and discrete. For any $A$-module $M$ and any homothety $f_r:M\to M,\ m\mapsto mr,\ r\in A$, if $\ker(f_r)=\{0\}$, ...