Flat embeddings of noetherian algebras in artinian rings (Q1802333)

If an algebra \(R\) embeds into an artinian ring \(S\) then the function \(\lambda\) on finitely generated right \(R\)-modules, taking values in \((1/n)\mathbb{Z}\) for some \(n\), defined by \(\lambda(M) = \text{length}(M\otimes_ RS)\) is easily seen to have the following properties: \(\lambda(R) = 1\), \(\lambda(R/I) < 1\) for each nonzero right ideal, \(\lambda(M\oplus N) = \lambda(M) + \lambda(N)\), and if \(N\) is a submodule of \(M\), \(\lambda(N) \leq \lambda(M) \leq \lambda(M/N) + \lambda(N)\). A remarkable theorem of Schofield states that the mere existence of such a function guarantees that there is an embedding of \(R\) into an artinian ring \(S\). Moreover, the embedding \(R \hookrightarrow{_ RS}\) is flat if and only if the final inequality above is replaced by equality. The author uses these results to prove various embedding results, generalizing results of Lenagan and Blair-Small. For example, he shows that a noetherian algebra \(R\) with finite Gelfand-Kirillov dimension and primary decomposition can be embedded in an artinian ring. In addition, he is able to give an explicit description of the prime middle annihilators of a noetherian algebra that has a flat artinian embedding.