Ideal intrinsic extensions with connections to PI-rings.
The paper under review studies two quite natural and important generalisations of PI rings (rings satisfying polynomial identities). These are as follows. A ring \(R\) is almost PI if every prime factor of \(R\) is PI. Moreover, \(R\) is intrinsically PI if each nonzero ideal of \(R\) contains a nonzero PI ideal. It is well known that if \(R\) is almost PI then the nonzero ideals of \(R\) have nontrivial intersection with \(\text{Cen}(R)\), the centre of \(R\). This comes to say that almost PI rings are, in a sense, a generalization to the semiprime PI rings. The authors call a ring \(R\) ideal intrinsic over its centre (IIC) if the nonzero ideals intersect \(\text{Cen}(R)\) nontrivially. (Note that sometimes such rings are called rings with large centres.) The paper under review studies, besides almost PI rings, also right bounded rings (that is, rings \(R\) such that each essential right ideal of \(R\) contains an ideal of \(R\) that is essential as a right ideal). First the authors prove that for a semiprime ring \(R\), PI implies almost PI, which in turn implies intrinsically PI. The latter implies also right bounded IIC and it implies IIC. Moreover, the implications are strict. Further on the authors study rings of quotients for intrinsically PI rings. They describe the behaviour of the condition IIC under ring extensions, and also radicals of IIC rings. It is shown that the set of the IIC ideals of a ring is closed under sums and intersections. This of course implies that every semiprime ring possesses a largest IIC ideal. Several additional properties of IIC rings relating them to semiprime ones are deduced as well.
scientific article; zbMATH DE number 5561728
scientific article