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?