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  "body": "\n## Summary\n\nTwo [[rings|0B1261B5ED]] are Morita equivalent when their categories of [[modules|0NZ8JA25WK]] are equivalent.\nSuch rings need not be isomorphic, but every property that is defined through modules alone is the same for both.\nA ring $R$ and the matrix ring $M_n(R)$ are the basic example.\nMorita theory describes all such equivalences through [[progenerators|ZQ9ABBKF1N]] and their endomorphism rings.\n\n## Explanations\n\n### L1 human\n\nOften one studies a ring through the modules over it, much as one studies a group through its actions.\nTwo different rings can have module theories that look exactly alike, with the same pattern of maps between modules.\nSuch rings are called Morita equivalent.\nFor example, the field $\\R$ and the ring of $2 \\times 2$ real matrices have the same module theory.\nA module over the matrix ring is a sum of copies of $\\R^2$, and these behave just like real vector spaces.\nSo any question about modules can be moved from one ring to the other.\n",
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