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  "body": "\n## Definition\n\nLet $P$ be a property of [[rings|0B1261B5ED]] with identity.\nThe property $P$ is a *Morita invariant* if the following holds.\nWhenever $R$ and $S$ are [[Morita equivalent|RD5WKA1VK7]] rings, $R$ has $P$ if and only if $S$ has $P$.\nMore 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.\n\n## Remarks\n\nExample: being left Noetherian is a Morita invariant.\nA ring is left Noetherian if and only if every finitely generated left module satisfies the ascending chain condition on submodules.\nFinite generation, submodules and chains of submodules are preserved by any equivalence of module categories.\nOther Morita invariant properties are being simple, semisimple, left Artinian, primitive, prime, left hereditary, and having a given left global dimension.\nThe [[center|MVT64B3B6H]] of a ring, the lattice of its two-sided ideals, and its Grothendieck group $K_0$ are Morita invariant objects.\nNon-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.\nFor the same reason, being a domain and being a local ring are not Morita invariants.\n",
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  "name": "Morita invariant",
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