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This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring      S [ θ  ; δ  ] S[\theta ;\delta ]      , where      S S       is a right noetherian ring with finite Krull dimension      n n       and      δ  \delta       is a derivation on      S S      . The main theorem states that      S [ θ  ; δ  ] S[\theta ;\delta ]       has Krull dimension      n n       unless there exists a simple right      S S      -module      A A       such that      A ⊗  S S [ θ  ; δ  ] A{ \otimes _S}S[\theta ;\delta ]       is not simple (as an      S [ θ  ; δ  ] S[\theta ;\delta ]      -module) and      A A       has height      n n       in the sense that there exist critical right      S S      -modules      A = A 0 , A 1 , …  , A n A = {A_0},{A_1},\ldots ,{A_n}       such that each      A i ⊗  s S [ θ  ; δ  ] {A_i} \otimes _s S[\theta ;\delta ]       is a critical      S [ θ  ; δ  ] S[\theta ;\delta ]      -module, each      A i {A_i}       is a minor subfactor of      A i + 1 {A_{i + 1}}       and      A n {A_n}       is a subfactor of      S S      . If such an      A A       does exist, then      S [ θ  ; δ  ] S[\theta ;\delta ]       has Krull dimension      n + 1 n + 1      . This criterion is simplified when      S S       is fully bounded, in which case it is shown that      S [ θ  ; δ  ] S[\theta ;\delta ]       has Krull dimension      n n       unless      S S       has a maximal ideal      M M       of height      n n       such that either      char( S / M ) &gt; 0 {\text {char(}}S/M) &gt; 0       or      δ  ( M ) ⊆  M \delta (M) \subseteq M      , and in these cases      S [ θ  ; δ  ] S[\theta ;\delta ]       has Krull dimension      n + 1 n + 1      .

Krull dimension of differential operator rings. III. Noncommutative coefficients