On almost FGP-injective modules

This paper gives the definition of AFGP-injective ring, we show: (1) If R is a semiprime right AFGP-injective ring, then every maximal right(or left) annihilator is a maximal right(left) ideal of R generated by an idempotent. (2) If R is a right AFGP-injective ring, and the ascending chain r(a
1
) ⊆ r(a
2
a
1
) ⊆ r(a
3
a
2
a
1
) ⊆ … terminates for every infinite sequence a
1
, a
2
, a
3
… of R. Then (a) R/Z(R
R
) is von Neumann regular. (b) R/J is tight T - nilpotent. (3)If R is a Baer ring, suppose for any a∈R, there exists 0 ≠ c ∈ R such that 0 ≠ ac = ca , and rl(ac) = acR⊕ X, then ac is a regular element of R.