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Very nearly \(\mathfrak C\) rings. (English) Zbl 1196.16001

Summary: Suppose \(C\subset R\) are rings with a common ideal \(A\neq 0\). Then they have similar properties: \(C\) is prime if and only if \(R\) is prime; if they are prime rings then \(C\) is PI if and only if \(R\) is PI, and they have the same PI class.
Also we can lift from \(C\) to \(R\) chains of prime ideals not containing \(A\), and we have the properties of LO, GU and INC which relate prime ideals of \(C\) not containing \(A\) with prime ideals of \(R\).
We want to check which other properties pass from \(R\) to \(C\) via common ideals. In a more general case \(C\) starts a chain of rings such that the last ring satisfies good properties, and each pair of consecutive rings has a common ideal.

MSC:

16D25 Ideals in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16S20 Centralizing and normalizing extensions
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
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