A counterpart to Nagata idealization

Idealization of a module K over a commutative ring S produces a ring having K as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an S-module K a subring R of S that behaves like the idealization of K but is such that when S is a domain, so is R. The ring S is contained in the normalization of R but is finite over R only when R=S. We determine conditions under which R is Noetherian, Cohen–Macaulay, Gorenstein, a complete intersection or a hypersurface. When R is local, then its m-adic completion is the idealization of the m-adic completions of S and K.