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Let $R$ be a noncommutative ring. The ring $R$ has descending chain condition on two-sided ideals (D.C.C.), if for 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 there any example of a ring $R$ with D.C.C. on two-sided ideals but without D.C.C on left ideals and also without D.C.C on right ideals? |
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Take the Weil algebra over a field $k$, i.e. $R=k\langle x,\frac{d}{dx}\rangle$ is the algebra with two generators, $x$ and $\frac{d}{dx}$, and one relation $\frac{d}{dx} x=1$. This algebra is well known to be simple, so it satisfies the DCC on two-sided ideals. For right ideals you have $xR\supsetneq x^2R\supsetneq x^3R\supsetneq\cdots$, and for left ideals $R\frac{d}{dx}\supsetneq R(\frac{d}{dx})^2\supsetneq R(\frac{d}{dx})^3\supsetneq\cdots$. |
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