When is a Sum of Annihilator Ideals an Annihilator Ideal?

Abstract

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.

Key Words:

• Annihilator-ideal
• Extremally disconnected space
• IN-ring
• Quasi-Baer ring
• SA-ring

2010 Mathematics Subject Classification:

• Primary 16D25
• 16D70
• 54G05

View addendum:

Corrigendum to: When is a sum of annihilator ideals an annihilator ideal?

ACKNOWLEDGMENT

We wish to thank the referee for her/his thorough reading of our paper and her/his comments which led to a much improved paper. The third author wishes to thank Professor A. R. Aliabad for his advice, particularly on the terminology for the class of SA-rings.

Notes

Communicated by T. Albu.