2 typos

# Chain CondictionCondition on Rings

Let $R$ be a noncommutative ring. The ring $R$ have 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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# Chain Condiction on Rings

Let $R$ be a noncommutative ring. The ring $R$ have 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?