PRIME RIGHT IDEALS AND RIGHT NOETHERIAN RINGS
1972; Elsevier BV; Linguagem: Inglês
10.1016/b978-0-12-291350-1.50019-x
Autores Tópico(s)Rings, Modules, and Algebras
ResumoThis chapter discusses the prime right ideals and right noetherian rings. A well-known theorem asserts that a commutative ring R is noetherian if every prime ideal of R is finitely generated. Using the definition of a prime right ideal, it is shown that Cohens theorem holds for any ring. The application of result gives an easy proof for the fact that the power series ring R[X, α] of the right noetherian ring R with the surjective endomorphism α is right noetherian. It is found that if every two-sided ideal of R and every prime right ideal of R is a finitely generated right ideal of R, then R is right noetherian. It is found that as every two-sided ideal of R is finitely generated, R satisfies the ascending chain condition on two-sided ideals. It is observed that if R is not right noetherian, then there is an ideal M of R that is maximal among the ideals X of R such that R/X is not right noetherian.
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