Embedding rings with Krull dimension in Artinian rings
An old problem in the theory of noetherian rings was the question whether they could be embedded into artinian rings. While a counterexample in the one-sided setting was presented by \textit{L. W. Small} in 1969 [Bull. Am. Math. Soc. 75, 147-148 (1969; Zbl 0185.09202)], the two-sided problem was not settled until nearly 20 years later, when \textit{C. Dean} and \textit{J. T. Stafford} constructed a counterexample [J. Algebra 115, No. 1, 175-181 (1988; Zbl 0641.16009)], \textit{C. Dean} then showed that the enveloping algebra of every finite dimensional, semisimple, complex Lie algebra has a factor ring which is not embeddable into an artinian ring [Bull. Lond. Math. Soc. 21, 65-69 (1989; Zbl 0671.17006)]. In the positive direction \textit{W. D. Blair} and \textit{L. W. Small} proved that a right noetherian algebra over a field, in which there are no nonzero right ideals with reduced rank zero, has a left flat embedding into an artinian ring [Isr. J. Math. 58, 10-18 (1987; Zbl 0626.16003)]; a converse, for two-sided noetherian algebras, was obtained by \textit{G. Krause} [Isr. J. Math. 77, No. 1-2, 97-114 (1992; Zbl 0780.16020)]. The present paper continues in the positive direction, outside the setting of algebras over a field. The main noetherian result is that if \(R\) is a noetherian ring, containing no nonzero right ideals with reduced rank 0, and if the clique of every associated prime ideal of \(R_R\) satisfies the incomparability condition, then \(R\) embeds in an artinian ring. A one-sided version of this theorem is also proved, in which the noetherian hypothesis is reduced to the assumption that \(R\) has right Krull dimension. Since there are no known examples of comparable prime ideals occurring in cliques in noetherian rings, the main result is quite general. The reduced rank condition is a natural one, as the authors show, in that it holds for any right noetherian ring which has a left flat embedding into a right artinian ring.
left flat embeddings into right Artinian rings
scientific article; zbMATH DE number 922320