Geodesic
Polynomial Modules Over Macaulay Modules
Canadian mathematical bulletin, Tome 19 (1976) no. 2, pp. 173-176

Voir la notice de l'article provenant de la source Cambridge University Press

In [2] we introduced a concept of a Macaulay module over a right noetherian ring by saying that all associated primes of the module have the same codimension. That is to say that a module M over a right noetherian ring R is Macaulay if K dim R/P = K dim R/Q for all P, Q ∈ Ass M. Our main aim here is to extend Nagata’s useful result [6], that Macaulay rings are stable under polynomial adjunction, to a noncommutative setting. Specifically, we prove where x is a commuting indeterminate, that the polynomial module M[x] = M⊗R R[x] is a Macaulay R[x]-module if and only if M is a Macaulay R-module. But actually, we prove a more general result. We show that when M is any module over a right noetherian ring, the associated primes of M[x] are precisely the extensions of the associated primes of M. 
Gordon, Robert. Polynomial Modules Over Macaulay Modules. Canadian mathematical bulletin, Tome 19 (1976) no. 2, pp. 173-176. doi: 10.4153/CMB-1976-026-0
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