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- W2500997462 abstract "A commutative ring R is said to be a φ-ring if its nilradical Nil(R) is both prime and divided, the latter meaning Nil(R) is comparable with each principal ideal of R. Special types include φ-Noetherian (also known as nonnilNoetherian), φ-Mori, φ-chained and φ-Prufer. A ring R is φ-Noetherian if Nil(R) is a divided prime and each ideal that properly contains Nil(R) is finitely generated. If R is a φ-Noetherian ring and X1,X2, . . . ,Xn are indeterminates, then an ideal I of R[X1,X2, . . . ,Xn] which contains a nonnil element of R is finitely generated. Also, for a ring R where Nil(R) is a nonzero prime ideal with Nil(R) = (0), there is a ring A whose nilradical Nil(A) is a divided prime such that R embeds naturally in A with R/Nil(R) isomorphic to A/Nil(A), RNil(R) isomorphic to ANil(A), and the corresponding total quotient rings, T (R) and T (A), are such that T (R) ⊂ T (A) and A ∩ T (R) = R+Nil(T (R))." @default.
- W2500997462 created "2016-08-23" @default.
- W2500997462 creator A5036537708 @default.
- W2500997462 date "2005-03-01" @default.
- W2500997462 modified "2023-09-26" @default.
- W2500997462 title "Rings with Prime Nilradical" @default.
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- W2500997462 doi "https://doi.org/10.1201/9781420028249.ch11" @default.
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