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TRANSCENDENTAL AND ALGEBRAIC EXTENSIONS OF COMMUTATIVE RINGS

 

作者: HJ Schutte,  

 

期刊: Quaestiones Mathematicae  (Taylor Available online 1988)
卷期: Volume 11, issue 1  

页码: 73-89

 

ISSN:1607-3606

 

年代: 1988

 

DOI:10.1080/16073606.1988.9631944

 

出版商: Taylor & Francis Group

 

关键词: Primary 13B02;Secondary 13B25

 

数据来源: Taylor

 

摘要:

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 aiover 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.

 

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