Polynomial Modules Over Macaulay Modules

Gordon, Robert

Canadian mathematical bulletin, Tome 19 (1976) no. 2, pp. 173-176

Résumé

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.

Détail

DOI : 10.4153/CMB-1976-026-0