A ringoid is a horizontal categorification of a ring. Since a ring is a monoid object in Ab, or equivalently a one-object category enriched over , a ringoid is equivalently a Ab-enriched category, often assumed to be small.
Ringoids share many of the properties of (noncommutative) rings. For instance, we can talk about (left and right) modules over a ringoid , which can be defined as Ab-enriched functors and . Bimodules over ringoids have a tensor product (the enriched tensor product of functors?) under which they form a bicategory, also known as the bicategory of -enriched profunctors. Modules over a ringoid also form an abelian category and thus have a derived category.
One interesting operation on ringoids is the (-enriched) Cauchy completion, which is the completion under finite direct sums and split idempotents. This is described in more detail at Ab-enriched category. In particular, the Cauchy completion of a ring is the category of finitely generated projective -modules (aka split subobjects of finite-rank free modules). Every ringoid is equivalent to its Cauchy completion in the bicategory , and two ringoids are equivalent in if and only if their Cauchy completions are equivalent as Ab-enriched categories. This sort of equivalence is naturally called Morita equivalence.
See also dg-category.