By Silvio Greco, Paolo Salmon (auth.)
The m-adic topologies and, particularly the notions of m-complete ring and m-completion A of a commutative ring A, ensue often in commutative algebra and also are a useful gizmo in algebraic geometry. the purpose of this paintings is to assemble jointly a few standards about the ascent (from A to A) and the descent (from A to A) of a number of houses of commutative earrings resembling, for instance: integrity, regularity, factoriality, normality, and so on. extra accurately, we wish to convey that a number of the above standards, even though no longer trivial in any respect, are effortless results of a few basic notions of commutative algebra and native algebra. occasionally we will get purely partial effects, which most likely could be greater through extra deeper investigations. No new consequence has been integrated during this paintings. Its purely origi nality is the alternative of fabric and the mode of presentation. The comprehension of crucial statements integrated during this ebook wishes just a very effortless historical past in algebra, excellent concept and normal topology. with the intention to emphasize the straightforward personality of our remedy, we've got recalled a number of popular definitions and, occasionally, even the proofs of the 1st homes which stick to at once from them. however, we didn't insert during this paintings a few vital effects, akin to the Cohen constitution theorem on whole noetherian neighborhood jewelry, as we didn't want to break out an excessive amount of from the spirit of the book.