We introduce in this paper the concept of left WMC2 rings and concern ourselves with rings containing an injective maximal left ideal. Some known results for left idempotent reflexive rings and left HI rings can be extended to left WMC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with nonzero socle and extend some known results on ...

In this paper, we shall study regular rings satisfying weak chain condition. As main results, we show that regular rings satisfying weak chain condition are unit-regular, and show that these rings have the unperforation and power cancellation properties for the family of finitely generated projective modules.

1. Flat, projective, and free modules 1 2. Review of regular local rings 2 3. Cohen-Macauley rings 2 4. The Koszul complex 3 5. The Koszul resolution detects depth 5 6. The detection of depth by use of Ext 6 7. Global dimension 6 8. Minimal resolutions and global dimension 7 9. Serre’s characterization of regular local rings 8 10. The Auslander-Buchsbaum theorem 10 11. Localizations of regular ...

It was pointed out in my last paper that there are rings whose real closure ∗ are not unique. In [4] we also discussed some example of rings by which there is a unique real closure ∗ (mainly the real closed rings). Now we want to determine more classes of rings by which real closure ∗ is unique. The main results involve characterisations of domains and Baer regular rings having unique real clos...

We use the concept of a regular object with respect to another object in an arbitrary category, defined in [3], in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories R−mod and S−mod, when S is an excellent extension of the ring R. Consequently, we obtain a result of [5]: if S is an excellent extension of the ring R, then S is von Neumann regular ring i...