Morita equivalence

In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. They are named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

Motivation

Rings are commonly studied in terms of their R-modules, as modules can be viewed as representations of rings. Every ring has a natural R-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via rings is more general and gives useful information. Because of this, one often studies a ring by studying the category of modules over that ring.

Morita equivalence takes this viewpoint to a natural conclusion by defining rings to be equivalent if their module categories are equivalent.

Formal definition

Two rings R and S are said to be Morita equivalent (or equivalent) if there is an additive equivalence of the category of (left) modules over R, _RM, and the category of (left) modules over S, _SM.

It can be shown that the left module categories are equivalent if and only if the right module categories are equivalent.

Equivalences can be characterized as follows: if F:_RM <math>\to</math> _SM and G:_SM <math>\to</math> _RM are additive (covariant) functors, then F and G are an equivalence if and only if there is a balanced (S,R)-bimodule P such that _SP and P_R are projective generators and natural isomorphisms <math> F \cong (P \otimes_R -)</math> and <math>G \cong Hom_R(P,-).</math>

Properties preserved by equivalence

Many properties are preserved by equivalence for the objects in the module category. Taking the ring as a special case, we have the following list of preserved properties between equivalent rings. If R and S are equivalent rings, then R is

• simple
• semisimple
• noetherian
• artinian
• primitive

if and only if S is. Furthermore, we have that Cen(R) is equivalent to Cen(S), where the Cen denotes the center of the ring, and R/J(R) is equivalent to S/J(S), where J denotes the Jacobson radical.

However, Morita equivalence is not equivalent to isomorphism. It is possible, but extremely difficult, to distinguish between non-isomorphic but Morita equivalent rings. (For example, via K-theory.) One important special case where Morita equivalence implies isomorphism is the case of commutative rings.

Examples

The ring of matrices with elements in R, M_n(R) is equivalent to R for any <math>n > 0</math>. Notice that this generalizes the classification of simple artinian rings given by Artin-Wedderburn theory.

Further directions

Dual to the theory of equivalences is the theory of dualities between the module categories, where the functors used are contravariant rather than covariant. This theory, though similar in form, has significant differences due to the fact that there is no duality between categories of modules for different rings, although dualities may exist for subcategories.

References

• F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2 nd Ed., Springer-Verlag, New York, 1992, ISBN 0387978453, ISBN 3540978453