# Chain Conditions on Ideals

Let $R$ be a noncommutative ring and $I$ a two-sided ideal of $R$. Assume that $I$ and $R/I$ both have descending chain condition on two-sided ideals (D.C.C.), that is, if we have a chain of two-sided ideals $J_1\supset J_2\supset \cdots$, then there exists an $N\in\mathbb N$ such that $J_n = J_N$ for $n\geq N$.

Is true that this implies that $R$ also verifies D.C.C. on two-sided ideals?

And for ascending chain condition on two-sided ideals (A.C.C) we have a similar result?

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Foe the A.C.C. case, you can read my answer here: math.stackexchange.com/a/118838/647 – Isaac Solomon Nov 22 '12 at 18:24

If $R$ is a ring and $0\rightarrow M^\prime\rightarrow M\rightarrow M^{\prime\prime}\rightarrow 0$ is an exact sequence of $R$-modules, then $M$ satisfies the ACC on $R$-submodules (resp. the DCC on $R$-submodules) if and only if both $M^\prime$ and $M^{\prime\prime}$ do. So if you apply this with $M^\prime=I$, $M=R$, and $M^{\prime\prime}=R/I$, you get what you want.
Also, I wouldn't say that $I$ satisfies the DCC or ACC on ideals, because $I$ is not a ring (unless $I=R$). It is also a fact that the $R$-submodules of $R/I$ are precisely the $R/I$-submodules, which are the ideals. This is why you can apply the result I mention above.