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Krull dimension of differential operator rings. III. Noncommutative coefficients

1983; American Mathematical Society; Volume: 275; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1983-0682736-5

1088-6850

K. R. Goodearl, T. H. Lenagan,

Algebraic structures and combinatorial models

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 ) > 0 {\text {char(}}S/M) > 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 .