Relaciones entre los grupos de Brauer de anillos conmutativos y ciertos anillos cocientes

Let S be a commutative R-algebra, R a commutative ring, then the application of R-algebras At → A ⨂R^S induces a homomorphism Br(R) → Br(S) between the Brauer groups of R and S, respectively. It is known that the homomorphism Br(R) →Br(R/m) is an isomorphism when R is a complete local ring with maximal ideal m. This article shows that Br(R) → Br(R/I) is an isomorphism whenever I is a nil-ideal and also when I is contained in the Jacobson radical of R and R is complete in the I-adic topology.