Near domains

Abstract

In this work a type of near ring that generalizes the ring theory concept of integral domain is defined and investigated. Such a near ring is called a near domain. O. Ore has shown that a non-commutative ring is a subring of a division ring if every pair of elements has a common left multiple. This is called the left Ore condition. A near domain is a left near ring that satisfies the right cancellation law and the left Ore condition. It is shown that a near domain is a C-ring, has no zero divisors and that left cancellation holds. Finite near domains are near fields. In fact, any finite near ring satisfying right cancellation is a near field so all proper near domains are infinite. Examples given show the existence of infinite near rings satisfying right cancellation but not the left Ore condition. Also, infinite near rings satisfying the left Ore condition and having no zero divisors but which are not near domains are exhibited. It is shown that a near domain can be embedded in a near field. This result is states as a corollary to a more general result showing constructability of near rings of left quotients with respect to a multiplicative set. If S is a multiplicative set of cancellable elements in a near ring R, then R has a bear ring of left quotients with respect to S if and only if the left Ore condition with respect to S holds. This is called the total near ring of left quotients if S is the set of all cancellable elements of R. An important consequence of the embedding corollary is the fact that addition in near domains must be commutative. This follows from B. H. Neumann's proof that the additive group of a near field is abelian. ...