Commutative algebra in subsystems of second order arithmetic

We determine which set existence axioms are needed to prove a number of well-known theorems of communicative algebra. We work in subsystems of second order arithmetic and we use the techniques of Reverse Mathematics.; We use the following four weak subsystems: RCA{dollar}sbsp{lcub}0{rcub}{lcub}*{rcub}{dollar}, RCA{dollar}sb0{dollar}, WKL{dollar}sb0{dollar}, ACA{dollar}sb0{dollar}. RCA{dollar}sb0{dollar} is a formal system that captures the notion of recursiveness. The axioms of WKL{dollar}sb0{dollar} are those of RCA{dollar}sb0{dollar} together with an axiom that states that every countable 0-1 tree has a path and the axioms of ACA{dollar}sb0{dollar} are those of RCA{dollar}sb0{dollar} together with an axiom scheme that guarantees the existence of sets that can be defined arithmetically. RCA{dollar}sbsp{lcub}0{rcub}{lcub}*{rcub}{dollar} is the weakest system we use. RCA{dollar}sbsp{lcub}0{rcub}{lcub}*{rcub}{dollar} together with the {dollar}Sigmasbsp{lcub}1{rcub}{lcub}0{rcub}{dollar} induction scheme is logically equivalent to RCA{dollar}sb0{dollar}. Our main results are: (1) Over the weak basis of RCA{dollar}sbsp{lcub}0{rcub}{lcub}*{rcub}{dollar}, the Fundamental structure for finitely generated abelian groups is equivalent to {dollar}Sigmasbsp{lcub}1{rcub}{lcub}0{rcub}{dollar} induction. (2) Over the weak basis of RCA{dollar}sb0{dollar}, a formal version of the Basis theorem for formal power series with coefficiencies from a countable commutative ring is equivalent to the existence of the ordinal number {dollar}omegaspomega{dollar}. (3) Over the weak basis of RCA{dollar}sb0{dollar}, the Extension theorem of valuations for countable fields is equivalent to Weak Konig's Lemma. (4) Over the weak basis of RCA{dollar}sb0{dollar}, Levi's theorem for countable abelian groups ("A countable abelian group is orderable if and only if it is torsion-free") is equivalent to Weak Konig's Lemma. (5) Over the weak basis of RCA{dollar}sb0{dollar}, the statement, "Every countable commutative ring has a minimal prime ideal," is equivalent to Arithmetic Comprehension.