Abstract
that this question is intimately connected with the existence of certain kinds of simple radical rings; it is this connection which we wish to emphasize in this paper. We define a left valuation ring to be a nonzero ring R in which, for every x, y in R, there exists a u in R such that either x = uy or y = ux. A right valuation ring is defined similarly. (A valuation ring in the sense of [1] above is a left and right valuation ring, in the sense of this paper, without divisors of zero.) There do exist left valuation rings which are not right valuation rings. An example can be given modifying [2, ?6, p. 219]. The construction is, however, omitted. We now prove some results valid for left valuation rings to lead up to the connection with simple radical rings. The lesser ones are called simply results, the more important ones theorems. RESULT 1. The left ideals of R are linearly ordered by inclusion.
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