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I am looking for an example of non-commutative rings $A$ such that $A^2 \cong A$ as left $A$-modules.

I am thinking about the simplest non-commutative ring $A=GL_n(\Bbb R)$ for $1 < n \in \Bbb Z_+$, could we find a suitable subspace of $A$?

Thank you for your help.

balddraz
  • 7,558
Conjecture
  • 3,088
  • 1
    There are two things wrong with your idea for an example: foremost, it is not a ring. I gather you probably meant the full ring of square matrices over a field. That still won’t work, nor any subring, because no Artinian ring can have that property. There are two similar standard examples discussed at the linked duplicates and extended links. – rschwieb Jul 26 '19 at 03:11
  • Right @rschwieb I meant the full ring, indeed $GL_n$ is not a ring. Thank you for the examples of dupes! – Conjecture Jul 26 '19 at 03:13

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