Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like R, S are Morita equivalent (denoted by ) if their categories of modules are additively equivalent (denoted by [a]).[2] It is 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 modules, as modules can be viewed as representations of rings. Every ring R has a natural R-module structure on itself where the module action is defined as the multiplication in the ring, so the approach via modules 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 Morita equivalent if their module categories are equivalent. This notion is of interest only when dealing with noncommutative rings, since it can be shown that two commutative rings are Morita equivalent if and only if they are isomorphic.
Definition
Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically
Examples
Any two isomorphic rings are Morita equivalent.
The ring of n-by-n
Criteria for equivalence
Equivalences can be characterized as follows: if F:R-Mod S-Mod and G:S-Mod R-Mod 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 PR are finitely generated projective generators and there are natural isomorphisms of the functors , and of the functors Finitely generated projective generators are also sometimes called progenerators for their module category.[3]
For every right-exact functor F from the category of left-R modules to the category of left-S modules that commutes with direct sums, a theorem of homological algebra shows that there is a (S,R)-bimodule E such that the functor is naturally isomorphic to the functor . Since equivalences are by necessity exact and commute with direct sums, this implies that R and S are Morita equivalent if and only if there are bimodules RMS and SNR such that as (R,R) bimodules and as (S,S) bimodules. Moreover, N and M are related via an (S,R) bimodule isomorphism: .
More concretely, two rings R and S are Morita equivalent if and only if for a
(isomorphism of rings) for some positive integer n and full idempotent e in the matrix ring Mn(R).
It is known that if R is Morita equivalent to S, then the ring Z(R) is isomorphic to the ring Z(S), where Z(-) denotes the
While isomorphic rings are Morita equivalent, Morita equivalent rings can be nonisomorphic. An easy example is that a division ring D is Morita equivalent to all of its matrix rings Mn(D), but cannot be isomorphic when n > 1. In the special case of commutative rings, Morita equivalent rings are actually isomorphic. This follows immediately from the comment above, for if R is Morita equivalent to S, .
Properties preserved by equivalence
Many properties are preserved by the equivalence functor for the objects in the module category. Generally speaking, any property of modules defined purely in terms of modules and their homomorphisms (and not to their underlying elements or ring) is a categorical property which will be preserved by the equivalence functor. For example, if F(-) is the equivalence functor from R-Mod to S-Mod, then the R module M has any of the following properties if and only if the S module F(M) does:
Many ring theoretic properties are stated in terms of their modules, and so these properties are preserved between Morita equivalent rings. Properties shared between equivalent rings are called Morita invariant properties. For example, a ring R is semisimple if and only if all of its modules are semisimple, and since semisimple modules are preserved under Morita equivalence, an equivalent ring S must also have all of its modules semisimple, and therefore be a semisimple ring itself.
Sometimes it is not immediately obvious why a property should be preserved. For example, using one standard definition of von Neumann regular ring (for all a in R, there exists x in R such that a = axa) it is not clear that an equivalent ring should also be von Neumann regular. However another formulation is: a ring is von Neumann regular if and only if all of its modules are flat. Since flatness is preserved across Morita equivalence, it is now clear that von Neumann regularity is Morita invariant.
The following properties are Morita invariant:
- semisimple
- von Neumann regular
- right (or left) Noetherian, right (or left) Artinian
- right (or left) self-injective
- quasi-Frobenius
- prime, right (or left) primitive, semiprime, semiprimitive
- right (or left) (semi-)hereditary
- right (or left) nonsingular
- right (or left) coherent
- semiperfect
- semilocal
Examples of properties which are not Morita invariant include
There are at least two other tests for determining whether or not a ring property is Morita invariant. An element e in a ring R is a full idempotent when e2 = e and ReR = R.
- is Morita invariant if and only if whenever a ring R satisfies , then so does eRe for every full idempotent e and so does every matrix ring Mn(R) for every positive integer n;
or
- is Morita invariant if and only if: for any ring R and full idempotent e in R, R satisfies if and only if the ring eRe satisfies .
Further directions
Dual to the theory of equivalences is the theory of
Morita equivalence can also be defined in more structured situations, such as for symplectic groupoids and C*-algebras. In the case of C*-algebras, a stronger type equivalence, called strong Morita equivalence, is needed to obtain results useful in applications, because of the additional structure of C*-algebras (coming from the involutive *-operation) and also because C*-algebras do not necessarily have an identity element.
Significance in K-theory
If two rings are Morita equivalent, there is an induced equivalence of the respective categories of projective modules since the Morita equivalences will preserve exact sequences (and hence projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of (roughly) the classifying space of the nerve of the (small) category of finitely generated projective modules over the ring, Morita equivalent rings must have isomorphic K-groups.
Notes
Citations
- ^ Anderson & Fuller 1992, p. 262, Sec. 22.
- ^ Anderson & Fuller 1992, p. 251, Definitions and Notations.
- ^ DeMeyer & Ingraham 1971, p. 6.
- ^ DeMeyer & Ingraham 1971, p. 16.
References
- Anderson, F.W.; Fuller, K.R. (1992). Rings and Categories of Modules. Zbl 0765.16001.
- DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics. Vol. 181. Berlin-Heidelberg-New York: Zbl 0215.36602.
- Zbl 0911.16001.
- Meyer, Ralf (1997). "Morita Equivalence In Algebra And Geometry" (PDF). CiteSeerX 10.1.1.35.3449. Retrieved August 22, 2023.
- Zbl 0080.25702.
Further reading
- Zbl 1024.16008.