Morita equivalence of two rings means equivalence of their categories of left modules.
For algebras over a specified commutative ring,
linear Morita equivalence requires the equivalence to respect that scalar action.
For example, a unital ring 
and its matrix ring 
, for positive integer 
, are Morita equivalent. The equivalence
identifies module-theoretic information without requiring
isomorphism of the rings.
Donovan's conjecture asks for finiteness of
Morita equivalence classes of blocks of finite group
algebras under a defect bound.
Morita Equivalence
See also
Block of a Group Algebra, Donovan Conjecture, Module, RingExplore with Wolfram|Alpha
References
Eaton, C. W.; Eisele, F.; and Livesey, M. "Donovan's Conjecture, Blocks with Abelian Defect Groups and Discrete Valuation Rings." Math. Z. 295, 249-264, 2020. https://doi.org/10.1007/s00209-019-02354-1.Etingof, P.; Golberg, O.; Hensel, S.; Liu, T.; Schwendner, A.; Vaintrob, D.; and Yudovina, E. Introduction to Representation Theory. Providence, RI: Amer. Math. Soc., 2011. https://doi.org/10.1090/stml/059.Sharifi, R. "Category Theory." Ch. 9 in Abstract Algebra, §9.3. https://www.math.ucla.edu/~sharifi/notes/algebra-ch09.html.Cite this as:
Weisstein, Eric W. "Morita Equivalence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MoritaEquivalence.html