Editing Ring theory (section)

===Morita equivalence===
{{main|Morita equivalence}}
Two rings ''R'', ''S'' are said to be [[Morita equivalent]] if the category of left modules over ''R'' is equivalent to the category of left modules over ''S''. In fact, two commutative rings which are Morita equivalent must be isomorphic, so the notion does not add anything new to the [[category theory|category]] of commutative rings. However, commutative rings can be Morita equivalent to noncommutative rings, so Morita equivalence is coarser than isomorphism. Morita equivalence is especially important in algebraic topology and functional analysis.