• SUM OF ANNIHILATOR NEAR-FIELD SPACES OVER NEAR-RING IS ANNIHILATOR NEAR-FIELD SPACE (SA-NFS-ONR-ANFS)

Abstract

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