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==For non-commutative rings== The Krull dimension of a module over a possibly non-commutative ring is defined as the [[deviation of a poset|deviation]] of the poset of submodules ordered by inclusion. For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.<ref>McConnell, J.C. and Robson, J.C. ''Noncommutative Noetherian Rings'' (2001). Amer. Math. Soc., Providence. Corollary 6.4.8.</ref> The two definitions can be different for commutative rings which are not Noetherian.
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