Associative rings and algebras Definition

Morita invariant

Definition

Let $P$ be a property of rings with identity. The property $P$ is a Morita invariant if the following holds. Whenever $R$ and $S$ are Morita equivalent rings, $R$ has $P$ if and only if $S$ has $P$ . More generally, an assignment of an object $X(R)$ to each ring $R$ is a Morita invariant if $X(R) \cong X(S)$ whenever $R$ and $S$ are Morita equivalent.

Remarks

Example: being left Noetherian is a Morita invariant. A ring is left Noetherian if and only if every finitely generated left module satisfies the ascending chain condition on submodules. Finite generation, submodules and chains of submodules are preserved by any equivalence of module categories. Other Morita invariant properties are being simple, semisimple, left Artinian, primitive, prime, left hereditary, and having a given left global dimension. The center of a ring, the lattice of its two-sided ideals, and its Grothendieck group $K_0$ are Morita invariant objects. Non-example: commutativity is not a Morita invariant, since a commutative ring $R \ne 0$ is Morita equivalent to $M_2(R)$ , which is not commutative. For the same reason, being a domain and being a local ring are not Morita invariants.

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Morita equivalence concept
Ring concept
Morita invariant definition

Tag	`G3NN0YEVVG`
URI	`m2://G3NN0YEVVG`
Kind	definition
Writing	W0 stub
MSC	16D90
Defines	Morita invariant
Uses	Ring, Morita equivalence
Changed	2 November 2026

Morita invariant

Definition

Remarks

Reading path

Further reading

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