12
votes
A question with simple and indecomposable modules
It's not true.
Consider representations of the quiver
$$\bullet\stackrel{\alpha}{\rightarrow}\bullet\stackrel{\beta}{\leftarrow}\bullet.$$
The representation
$k \to k^2
\leftarrow k$, where the ...
- 35.2k
8
votes
A question with simple and indecomposable modules
As shown by Jeremy Rickard's answer, $S := S_0$ is usually not contained in an indecomposable direct summand. The purpose of this answer is to show the weaker statement
$S$ can be embedded into an ...
- 2,160
8
votes
Accepted
commutative, infinite, artinian ring (with unity) in which distinct ideals has distinct index
Let $R$ be an infinite, commutative, Artinian ring with the distinct-index property for distinct ideals. The claim is that $R$ is a field.
Case 1. $R$ has a nonzero ideal $I$ such that $I^2=0$.
...
- 14.6k
6
votes
Accepted
Structure theorem for artinian modules?
No, in general local finite dimensional commutative $K$-algebras are wild and their modules can not be classified. For example when $m$ is the maximal ideal and $m/m^2$ has $K$-dimension at least 3, ...
- 26.5k
5
votes
Accepted
Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$
We use the fact that in an Artinian Gorenstein ring, any ideal contains the socle. The assumption tells us that the socle of $A$ is $\mathfrak m^2$, which is principal.
Let $I\neq (0)$ be a non-...
- 30.5k
4
votes
Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$
The answer by @Hailong Dao can be slightly generalized to easily show the following:
Let $(R,\mathfrak m,k) $ be an Artinian local Gorenstein ring with $\mathfrak m^3=0 $ and $\mathfrak m^2\ne 0.$
...
- 642
4
votes
Accepted
injective hull and projective cover of simple modules are indecomposable
One definition of "projective cover" of $S$ is that it is a projective module $P$, together with an epimorphism $\phi\colon P\to S$ such that the kernel $K$ is a superfluous submodule of $P$,...
- 13k
4
votes
Artinian Gorenstein subrings with same socle degree
Let $A=k[x,y]/(x^2,y^2)$. If $\mathrm{char}(k)\neq 2$, then the
subalgebra generated by $x+y$ is Gorenstein with the same socle
degree. On the other hand, for any graded Artinian algebra $A$, any
...
- 50.8k
3
votes
Structure theorem for artinian modules?
We shouldn't expect a general classification theorem for modules. Take $A=K[x,y]/(x^2,y^2)$. If $K$ is characteristic 2 this is the group algebra for the Klein 4 group.
There are infinitely many ...
- 532
2
votes
Accepted
On the finiteness of an Auslander-Reiten component
Since $A$ is an Artin algebra, it has only finitely many indecomposable injective modules in total (up to isomorphism), so there are finitely many in $\Gamma$.
In a locally finite quiver, given any $d\...
- 1,690
1
vote
Rings of finite uniserial type
Only not to leave unanswered the well known and easy part of the question, i.e.
especially in the case that the Artinian ring is also a principal ideal ring (two-sided Artinian and two-sided ...
- 486
1
vote
Accepted
Rings of finite uniserial type
It is an open problem which Artin algebras have only finitely many uniserial indecomposable modules.
This is stated for example as problem 2 in the open problems section in the book "...
- 26.5k
1
vote
On Artinian rings
I found the answer of the above question. The answer is this: Under this assumption R does not need to be right Artinian. Thank you.
- 321
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