ON INJECTIVITY AND P-INJECTIVITY, IV

Title & Authors
ON INJECTIVITY AND P-INJECTIVITY, IV
Chi Ming, Roger Yue;

Abstract
This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $\small{_{A}}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $\small{_{A}}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).
Keywords
Von Neumann regular;self injective rings;p-injectivity;YJ-injectivity;
Language
English
Cited by
1.
A Note on GP-Injectivity, Algebra Colloquium, 2009, 16, 04, 625
References
1.
Rend. Sem. Mat. Univ. Padova, vol.72. pp.117-133

2.
Comm. Algebra, vol.23. pp.841-861

3.
AMS Math. Surveys and Monographs, vol.65.

4.
Ring Theory : Nonsingular rings and modules,

5.
Von Neumann regular rings,

6.
Publ. Math., vol.38. pp.455-461

7.
London Math. Soc. Monographs, vol.17. C.U.P.,

8.

9.
London Math. Soc. Lecture Note Series, vol.147. C.U.P.,

10.
Journal of Algebra, vol.174. pp.77-93

11.
Glasgow Math. J., vol.37. pp.373-378

12.
Foundations of module and ring theory,

13.
Riv. Mat. Univ. Parma, vol.1. 6, pp.31-37

14.
Math. J. Okayama Univ., vol.28. pp.133-146

15.
Glasgow Math. J., vol.37. pp.21-31

16.
Proc. Edinburgh Math. Soc., vol.19. pp.89-91

17.
Math. Japonica, vol.19. pp.173-176

18.
Math. Scandinavica, vol.39. pp.167-170

19.
Math. J. Okayama Univ., vol.20. pp.123-129

20.
Rend. Sem. Mat. Univ. Torino, vol.39. pp.75-84

21.
Riv. Mat. Univ. Parma, vol.8. 4, pp.443-452

22.
Glasnik Mat., vol.18. 38, pp.221-229

23.
Annali di Mat., vol.138. pp.245-253

24.
Riv. Mat. Univ. Parma, vol.11. 4, pp.101-109

25.
Ann. Univ. Fenara, vol.31. pp.49-61

26.
J. Math. Kyoto Uni., vol.27. pp.439-452

27.
Acta Math. Vietnamica, vol.13. pp.71-79

28.
Riv. Mat. Univ. Parma, vol.4. 6,

29.