The purity of a completion

S. H. Cox
1976 Glasgow Mathematical Journal  
This note establishes two statements from R. M. Fossum's review [1] of a paper by E. A. Magarian [2] . Firstly, if A -» B is a pure homomorphism (of commutative rings) then A [[x u ..., x s ]]-*B [[x lt ...,x s ]] is pure. Secondly, if R n -*R is a directed family of pure homomorphisms then \jR n -+ R is pure. A consequence is that if R n -> R is a directed family of pure homomorphisms and if R is Noetherian, then uR n [[x t , ..., x s ]] is Noetherian. A homomorphism A -> B is said to be pure
more » ... respectively n-pure) if for every ,4-module M (respectively generated by n elements) the natural bimodule map M-+M®B is injective. Clearly a morphism is pure if and only if it is n-pure for each n. The notion of n-pure is equivalent to what Gilmer and Mott [3] called condition £", namely each system of n linear equations over A which has a solution in B already has a solution in A. This equivalence is easily seen by observing that each of n-pure and M -> 0 and matrix representing / . Define M, = coker {R'" -*'Rf) if the elements of the matrix of/are in R t and M, = 0 otherwise. For each s 2: < let M, -> M s be the natural bimodule homomorphism obtained by extension of scalars R, -* R s . Then lim M t = M and M t is generated by n elements over R t . Hence M, -*• M t ® Rt R is injective for each t; the injectivity of M -> M® A R follows from the exactness of direct limits. https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s0017089500002846 fatcat:7426mflswfdvxj4mu3tcotafoa