Embedding Noetherian rings in Artinian rings (Q5956266)

A well-known theorem of \textit{A. H. Schofield} [``Representation of rings over skew fields'', Lond. Math. Soc. Lect. Note Ser. 92, CUP, Cambridge (1985; Zbl 0571.16001)] asserts that an algebra \(A\) over a field can be embedded in a right Artinian ring if and only if there is a faithful Sylvester rank function on finitely presented \(A\)-modules. By \textit{C. Dean} and \textit{J. T. Stafford} [J. Algebra 115, No. 1, 175-181 (1988; Zbl 0641.16009)], Schofield's result was used to show that a certain factor of the enveloping algebra of \(\text{sl}(2,\mathbb{C})\) could not be embedded in an Artinian ring. The non-existence of a rank function being quite difficult to establish, the main purpose of the present paper is to give an easily-checked criterion for embeddability which applies to the \(\text{sl}(2,\mathbb{C})\) example. Namely, it is shown that if \(R\) is an irreducible Noetherian ring with a non-zero right ideal with reduced rank \(0\), and the nilpotent radical \(N\) of \(R\) does not contain its left annihilator \(l(N)\), then \(R\) is not embeddable in a right Artinian ring.