Factor Rings and Splitting Ideals of Self-Injective, CS and Baer Rings

ABSTRACT

We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5).

These results also generalize results of the author in Faith (1985), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring.

The results are applied in Section 6 to extend theorems of Faith (1996) characterizing VNR rings, and, as the title of Faith (1996) suggests, extend the conjecture of Shamsuddin.

Keywords:

• Annihilator right ideal
• Baer ring
• Complement right ideal
• Co-nonsingular ring
• CS-ring
• Dedekind Finite ring
• FP-injective
• Ideal
• Nonsingular ring
• p-injective ring
• Prime ideal
• Prime ring
• Regular ideal
• Right annulet
• Ring
• Self-injective ring
• Semiprime ideal
• Semiprime ring
• von Neumann regular ring
• Quasi-
• or pseudo-Frobenius ring

Mathematics Subject Classification:

• Primary 16D50
• 16D60
• 16E50
• 16L60
• 16N60
• 16S90

Notes

^#Communicated by A. Facchini.