• DENSELY CO-HOPFIAN SUB NEAR-FIELD SPACES OVER NEAR-FIELD

Abstract

The sub near-field spaces of near-field of the title are introduced as a proper generalization of quasi co-Hopfian sub near-field spaces of near-field, and are characterized in several ways. The near-field space N is right non-singular if and only if the densely co-Hopfian right N-sub near-field spaces over near-field and quasi co-Hopfian right N-near-field spaces over near-field coincide. Dense co-Hopficity is investigated for certain N-sub near-field spaces over near-field that has indecomposable decompositions complementing direct summands. For some classes of near-field spaces N, including rings, near-rings, regular d-near-rings, near-fields with dense right socles, we determine when the injective envelope of an N-sub near-field space over near-field is densely co-Hopfian.