Krull symmetric Noetherian rings and their homogeneous components. (English) Zbl 0591.16013
A noetherian ring R is Krull symmetric if for every pair of ideals \(T\subseteq S\) of R the right and left Krull dimensions of S/T are equal. A module M is K-homogeneous if all non-zero submodules of M have the same Krull dimension. In general, the sum of two homogeneous left ideals of the same Krull dimension need not be homogeneous. However, if R is a Krull symmetric noetherian ring such that the associated primes of R are minimal primes, then the sum of homogeneous left ideals with the same Krull dimension is homogeneous. In this case, for any ordinal \(\alpha\), the ring contains a unique maximal homogeneous ideal \(H_{\alpha}\) of Krull dimension \(\alpha\). Only finitely many \(H_{\alpha}\) are non-zero and their direct sum is a faithful ideal of R. The ring R has an artinian classical quotient ring if and only if this non-zero ideal contains a non-zero-divisor. Finally, the previous result is used to obtain another proof of a theorem due to Warfield concerning decompositions of rings.
Reviewer: T.H.Lenagan
MSC:
16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |
16P40 | Noetherian rings and modules (associative rings and algebras) |
16P50 | Localization and associative Noetherian rings |
16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
16Dxx | Modules, bimodules and ideals in associative algebras |
Keywords:
right and left Krull dimensions; homogeneous left ideals; Krull symmetric noetherian ring; minimal primes; sum of homogeneous left ideals; maximal homogeneous ideal; faithful ideal; artinian classical quotient ring; decompositions of ringsReferences:
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