Vol. 25, No. 1, 1968

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A radical for lattice-ordered rings

John Edwin Diem

Vol. 25 (1968), No. 1, 71–82

The main result of this paper states that for a lattice-ordered ring (l-ring) A with no nonzero nilpotent l-ideals the following are equivalent: (i) A is an f-ring; (ii) A is a subdirect union of totally-ordered rings with no nonzero divisors of zero; (iii) x+x = 0 for all x A; (iv) x+ax = 0 for all x,a A; and (v) a(b c) = ab ac and (b c)a = ba ca for all a,b,c A with a 0. In particular, the equivalence of (i) and (iii) implies that an l-ring which has an identity that is a weak order unit and which has no nonzero nilpotent l-ideals is necessarily an f-ring.

The basic tool in our considerations is the notion of prime l-ideal. Specifically, call a proper l-ideal P of an l-ring A prime if I P or J P wherever I and J are l-ideals of A with IJ P. Various conditions are obtained on A, each of which forces A modulo every prime l-ideal to be totally-ordered with no nonzero divisors of zero. Moreover the relationship between the join of all the nilpotent l-ideals of A and the intersection of all the prime l-ideals of A is investigated in order to obtain the theorem mentioned above.

Mathematical Subject Classification
Primary: 06.85
Received: 9 January 1967
Published: 1 April 1968
John Edwin Diem