TRANSCENDENTAL AND ALGEBRAIC EXTENSIONS OF COMMUTATIVE RINGS

Abstract

Abstract Transcendental and algebraic elements over commutative rings are defined. Rings with zero nil radical are considered. For a transcendental over R, necessary and sufficient conditions are derived for elements of R[α] to be algebraic or transcendental over R. For R a ring with identity and a finite number of minimal prime ideals, necessary and sufficient conditions are given for any element in a unitary overring of R to be algebraic or transcendental over R. It is proved that if α is algebraic Over R, so is every element of R[α]. It is show that if R is Noetherian, β is algebraic over R[α] and α is algebraic over R, then, under certain conditions, β is algebraic over R. If R has a finite number of minimal prime ideals, P1,…,Pk, which are pairwise comaximal, then if t is transcendental over R, R[t] can be obtained by adjoining k algebraic elements ai over R to R whose defining polynomials are in Pi [x], and conversely, if such elements are adjoined to R, they generate an element transcendental over R.