Every commutative perfect ring is a quotient of an almost perfect domain

Abstract

In this paper, we show that every nonzero commutative perfect ring is a quotient of a one-dimensional semilocal almost perfect domain of characteristic zero. The domain can be chosen to have as many maximal ideals as the given ring has local factors. The construction uses coefficient discrete valuation rings, triangular presentations of purely inseparable residue extensions, and approximation for finitely many independent rank-one valuations. Independent error variables keep the construction inside a field. A normal-form argument shows that the divided elements used to get perfect principal quotients do not kill any further good monomial. In positive residue characteristic, the choice $a=2c+1$ with $c\ge n$ protects the nonzero powers of $p$ in a target ring of characteristic $p^n$. We also give local and mixed-characteristic consequences, and examples with non-nilpotent radicals.