On a retract of an integral domain

Abstract

Let A ⊂R be commutative integral domains with identity. We say A is a retract of R when there exists a ring homomorphism τ: R→A such that τ|_A=id_A. We Prove two theorems: Theorem 3.1 says when A is a retract of R via τ, with Kernel (τ)= =t·R≠ (O), then t is transcendental over A and if A is “sub-inert” in R, then R∩K(t)=A[t], where K is the quotient field of A. This theorem with the hypotheses strengthened to A being a “2-valuation algebra” of R and the transcendence degree of R over A equal to one, gives Theorem 4.1, which now says R=A[t]. Given the other hypotheses of Theorem 4.1, A being a “2-valuation algebra” of R is equivalent to the existence of a valuation ring V of L, the quotient field of R, such that V∩R=A. Some counterexamples are given and some variations on the hypotheses of Theorem 4.1 are discussed.