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Two rings are said to be Morita equivalent if their categories of (left) modules are equivalent. The notion is also used in more general contexts when certain categories of representations are equivalent.
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Does Morita theory hint higher modules for noncommutative ring?
As explained for example here
https://doi.org/10.1112/plms/s3-72.2.281
the most direct analogue of Morita equivalence for (connected) graded algebras actually boils down to graded isomorphism.
However …