Why is $M/(N\cap P)$ Artinian when $M/N$ and $M/P$ are Artinian?

  • MHB
  • Thread starter Euge
  • Start date
  • Tags
    2016
In summary, an Artinian module is one that satisfies the descending chain condition. This relates to the question about $M/(N\cap P)$ as the quotient module must also be Artinian if both $M/N$ and $M/P$ are Artinian. The Artinian property is preserved under quotients because the submodules of the quotient correspond to submodules of the original module. An example of this is seen with $M/(N\cap P)$ where $M/N$ and $M/P$ are isomorphic to $\mathbb{Z}/3\mathbb{Z}$ and $M/(N\cap P)$ is isomorphic to $\mathbb{Z}/2\mathbb{
  • #1
Euge
Gold Member
MHB
POTW Director
2,073
244
Here is this week's POTW:

-----
Let $R$ be a commutative ring. If $N$ and $P$ are submodules of an $R$-module $M$ such that $M/N$ and $M/P$ are Artinian, show that $M/(N\cap P)$ is Artinian.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
  • #2
No one answered this week's problem. You can read my solution below.
$M/(N\cap P)$ is isomorphic to a submodule of the Artinian module $M/N \times M/P$ via the $R$-mapping $M/(N\cap P) \to M/N \times M/P$ given by $m + N\cap P \mapsto (m + N, m + P)$; hence, $M/(N\cap P)$ is Artinian.
 

FAQ: Why is $M/(N\cap P)$ Artinian when $M/N$ and $M/P$ are Artinian?

What is the definition of an Artinian module?

An Artinian module is a module that satisfies the descending chain condition, meaning that any decreasing chain of submodules eventually stabilizes.

How does the definition of Artinian modules relate to the question about $M/(N\cap P)$?

The question is asking about the Artinian property of the quotient module $M/(N\cap P)$. In order for this quotient to be Artinian, both $M/N$ and $M/P$ must also be Artinian.

Why is the Artinian property preserved under quotients?

The Artinian property is preserved under quotients because if a module is Artinian, then any decreasing chain of submodules of the quotient will also eventually stabilize, since the submodules of the quotient correspond to submodules of the original module.

Can you provide an example to illustrate why $M/(N\cap P)$ is Artinian when $M/N$ and $M/P$ are Artinian?

Consider the module $M=\mathbb{Z}/6\mathbb{Z}$, and let $N=2\mathbb{Z}/6\mathbb{Z}$ and $P=3\mathbb{Z}/6\mathbb{Z}$. Both $M/N$ and $M/P$ are isomorphic to $\mathbb{Z}/3\mathbb{Z}$, which is Artinian. Then, $M/(N\cap P)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, which is also Artinian. This example shows that the Artinian property is preserved under quotients.

How does the Artinian property of $M/(N\cap P)$ relate to the submodules $N$ and $P$?

The Artinian property of $M/(N\cap P)$ implies that $N\cap P$ is a finite intersection of Artinian submodules of $M$. This means that $N$ and $P$ must also be Artinian, since any finite intersection of Artinian submodules is itself Artinian.

Similar threads

MHB How Do You Solve This Integral for Positive Integer n?
Replies
1
Views
4K
MHB Can Even Maps Between n-Spheres Have Odd Homological Degrees?
Replies
1
Views
2K
MHB Is every $S^{-1}R$-module flat?
Replies
1
Views
2K
MHB What is the maximal function of a finite Borel measure on $\Bbb R^n$?
Replies
1
Views
2K
MHB Why are nondegenerate fixed points important for self-maps of a smooth manifold?
Replies
1
Views
2K
MHB What Is the Center of the Ring of All \( n \times n \) Complex Matrices?
Replies
1
Views
5K
MHB What is the sum of quadratic residues in $\Bbb Z/p\Bbb Z$ for a prime $p > 3$?
Replies
1
Views
2K
MHB Is the Tensor Algebra of $\Bbb Z/n\Bbb Z$ Isomorphic to $\Bbb Z[x]/(nx)$?
Replies
1
Views
2K
MHB Is every polynomial a unit in a commutative ring with unity?
Replies
1
Views
2K
MHB Can a Function Belong to L^1 and have a Maximal Function in L^1?
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top