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We know that a non-commutative ring may have different numbers of left ideals and two-sided ideals. For example, a matrix ring over a field has only 2 two-sided ideals but it have some non-trivial left ideals. My question is about the number of left ideals and right ideals. Does every non-commutative ring have the same numbers of left ideals and right ideals?

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  • $\begingroup$ What about the ring of upper triangular matrices? $\endgroup$ – N. S. Oct 13 '12 at 3:26
  • $\begingroup$ Crossposted: mathoverflow.net/questions/109510/… $\endgroup$ – Qiaochu Yuan Oct 13 '12 at 3:43
  • $\begingroup$ @N.S. that ring is isomorphic to its opposite ring, so it has the same number of left and right ideals. $\endgroup$ – Mariano Suárez-Álvarez Oct 13 '12 at 4:09
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For another example, see the skew polynomial ring constructed in this solution: https://math.stackexchange.com/a/146406/29335

Given a field endomorphism $\rho: F\rightarrow F$ such that $[F:\rho(F)]\geq 2$, it produces a ring with exactly three left ideals and at least four right ideals.

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Consider the subring $R$ of the matrix ring $M_2(\mathbb R)$ of all matrices of the form $\begin{pmatrix}a&b\\0&c\end{pmatrix}$ with $a\in\mathbb Q$, and $b$, $c\in\mathbb R$.

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