Idempotent lifting and ring extensions. (Q2803566)

Whether a specific property lifts from a factor ring back to the ring is a common problem investigated for various ring properties. Knowing if it does is useful; often having many other salient properties following from it. As indicated by the title, in this paper the authors look at lifting of idempotents, i.e. if \(I\) is an ideal of a ring \(R\) and \(x+I\) is idempotent in \(R/I\), then there is an idempotent \(e\) of the ring \(R\) with \(x+I=e+I\). This problem is of course not original, but still there are many interesting and open questions related to it and here the authors answers many of these. We mention a few.NEWLINENEWLINE The first main result is that the lifting of idempotents modulo the Jacobson radical is not Morita invariant. In particular, by giving a suitable example, it is shown that the Jacobson radical of a ring can lift an idempotent, but the Jacobson radical of the corresponding matrix ring does not.NEWLINENEWLINE A notion related to the lifting of idempotents is that of an enabling ideal. An ideal \(I\) of a ring \(R\) is enabling in \(R\) if, whenever \(x+I=e+I\) for some elements \(e\) and \(x\) in \(R\) with \(e\) idempotent in \(R\), then there is an idempotent \(f\) in \(xR\) with \(e+I=f+I\). It is known that the Jacobson radical is always enabling. One of the reasons for considering these ideals is because the sum of two enabling ideals is enabling which is not the case for the sum of two ideals that lift idempotents.NEWLINENEWLINE It is shown that enabling does not necessarily transfer to polynomials (if \(I \) is an enabling ideal of \(R\), then \(I[t]\) need not be enabling in \(R[t]\)), but it does for power series rings (if \(I\) is an enabling ideal of \(R\), then \(I[[t]]\) is enabling in \(R[[t]]\)).NEWLINENEWLINE The paper concludes with a number of open problems; one of which is whether enabling is Morita invariant.