### TAG-modules with complement submodules $h$-pure.

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I. S. Cohen proved that any commutative local noetherian ring R that is J(R)-adic complete admits a coefficient subring. Analogous to the concept of a coefficient subring is the concept of an inertial subring of an algebra A over a commutative ring K. In case K is a Hensel ring and the module ${A}_{K}$ is finitely generated, under some additional conditions, as proved by Azumaya, A admits an inertial subring. In this paper the question of existence of an inertial subring in a locally finite algebra is discussed....

A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that Ā = A/J(A) is a separable field extension of R̅ = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R₀ which is a chain ring such that A = R₀ + J(A) and R₀ ∩ J(A) = J(R₀) = J(R)R₀. The structure of A in terms of a skew polynomial ring over R₀ is determined.

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