Morita Equivalence - Properties Preserved By Equivalence

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: injective, projective, flat, faithful, simple, semisimple, finitely generated, finitely presented, Artinian, and Noetherian. Examples of properties not necessarily preserved include being free, and being cyclic.

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:

• simple, 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
• semiprimary, right (or left)perfect, semiperfect
• semilocal

Examples of properties which are not Morita invariant include commutative, local, reduced, domain, right (or left) Goldie, Frobenius, invariant basis number, and Dedekind finite.

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 M_n(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 .