  This article mentions that Smith ideal theory generalizes the adic completion
theory of commutative rings to monoid objects of locally presentable symmetric
monoidal abelian categories. As an application, we provide an almost
mathematics version of completion theory, and we prove that the project limit
of the residue algebras by powers of almost finitely generated ideals of almost
algebras are complete as well-known results of commutative algebra theory.
  Furthermore, we introduce a naive construction of the completion theory of
$\mathbb{E}_\infty$-rings, which is a homotopy theory analogue of the adic
completion theory of commutative algebra by Hovey's Smith ideal theory. We
prove that the completion is homotopically complete for any weakly compact
Smith ideal as the ordinal commutative algebra theory. Furthermore, by using
the completion theory of Smith ideals of the symmetric monoidal category of
motivic spectra due to Voevodsky, we formulate an approximation of algebraic
cobordism by algebraic $K$-theory, and we prove that the approximation inherits
properties of algebraic $K$-theory, like the Bott periodicity and the Gabber
rigidity.
