• INFINITE SUB-NEAR-FIELDS OF INFINITE NEAR-FIELDS AND NEAR LEFT ALMOST – NEAR- FIELDS (IS-NF-INF-NLA-NF)

Abstract

In this paper, I studied and obtain some results on every infinite associative near-field contains an infinite commutative sub-near-field, and thereby suggested the problem of finding reasonably small classes  of infinite near-fields with the property that every infinite near-field contains a sub-near-field belonging to . Clearly, there is no minimal class  in the obvious sense, for in any class satisfying a near-field may be replaced by any proper infinite sub-near-field of itself. We determine a class 0 satisfying and consisting of familiar and easily-described zero symmetric near-fields; and we indicate how my results subsume and extend known finiteness results formulated in terms of sub-near-fields and zero divisors.

In last section identifies classes which satisfy and are minimal in a certain loose sense, and it extends the major result of the other sections to distributive near left almost near-fields. The field-theoretic results are proved in the setting of the alternative near-fields.