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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 $K$ is any left ideal of $R$?

Also, is $I/J$ is torsion? We get finitely generated for free because of Noetherian-ness, I believe.

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First of all, for $I\subseteq J$, it's usually customary to write the quotient as $J/I$ rather than $I/J$. Secondly, what do you mean by torsion? Most sources only talk about torsion modules over a domain, but there are other definitions... Finally, what does being finitely generated have to do with being torsion? (I guess this depends on your definition of torsion.) –  rschwieb Oct 10 '12 at 12:22
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Can the factor ring $I/J$ be expressed in terms of sums, quotients or submodules of rings of the form $R/K$, where $K$ is any left ideal of $R$?

The most sensible correction to this I can see is

Can the factor ring $J/I$ be expressed in terms of sums, quotients or submodules of modules of the form $R/K$, where $K$ is a left ideal of $R$?

Notice that I had to switch "rings" to "modules" because you required $K$ to be a left ideal, but $R/K$ won't be a ring unless it is a two-sided ideal.

In this interpretation, $J/I$ is a submodule of $R/I$.

It's unclear what definition of torsion you wanted, but you haven't clarified for over a year, so I'll give it a shot.

Certainly $J/I$ does not have to be a singular module, and it certainly might have a nonzero element with annihilator zero, so it would not be torsion in either of these senses.

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Surely you mean to consider the factor ring $J/I$.

You have $J/I"="\frac{R}{I}*\frac{I}{R}"="(R/I)/(R/J)$.

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$(R/I)/(R/J)$ doesn't make any sense. I think you're thinking of $(R/J)/(I/J)\cong R/I$, but I could have misunderstood. –  rschwieb Nov 7 '13 at 18:42
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