## $$l$$-reduced lattice-ordered rings with maximum condition on $$l$$-annihilators.(English)Zbl 1012.06021

The author describes the structure of $$l$$-reduced lattice-ordered rings with maximum conditions on $$l$$-annihilators.
Let $$R$$ be a lattice-ordered ring ($$l$$-ring) and $$S$$ be a non-empty subset of $$R$$; then the right $$l$$-annihilator $$S_r$$ of $$S$$ is defined to be the set $$\{ x\in R \mid |s||x|>0, \text{ for all } s\in S\}$$. The left $$l$$-annihilator $$S_l$$ of $$S$$ is defined analogously. If $$R$$ is an $$l$$-reduced $$l$$-ring then $$S_r=S_l$$ is a two-sided $$l$$-ideal; it is also called $$l$$-annihilator of $$S$$ and is denoted by $$S^*$$.
The author proves that for an $$l$$-reduced $$l$$-ring $$R$$ and for a non-empty subset $$S$$ of $$R$$ the following conditions are equivalent: (a) $$S^*$$ is a maximal $$l$$-annihilator; (b) $$S^*$$ is an $$l$$-prime $$l$$-ideal; (c) $$S^*$$ is a maximal $$l$$-prime $$l$$-ideal; (d) $$S^*$$ is a completely $$l$$-prime $$l$$-ideal; (e) $$S^*$$ is a minimal completely $$l$$-prime $$l$$-ideal.
The main result of the paper is the following.
For any $$l$$-ring $$R$$ the following statements are equivalent: (a) $$R$$ is $$l$$-reduced and has the maximum condition on $$l$$-annihilators; (b) $$R$$ is $$l$$-reduced and has only a finite number of maximal $$l$$-annihilators, $$M_1, M_2, \dots , M_s$$, satisfying $$\bigcap_{i=1}^sM_i=\{ 0\}$$; (c) $$R$$ is $$l$$-reduced and has only a finite number of distinct minimal $$l$$-prime $$l$$-ideals, $$P_1, P_2, \dots , P_n$$, satisfying $$\bigcap_{i=1}^nP_i=\{0\}$$; (d) $$R$$ is $$l$$-reduced and has only a finite number of distinct minimal completely $$l$$-prime $$l$$-ideals, $$Q_1, Q_2, \dots, Q_m$$, satisfying $$\bigcap_{i=1}^mQ_i=\{ 0\}$$; (e) $$R$$ is isomorphic to a subdirect sum of finitely many $$l$$-domains.

### MSC:

 06F25 Ordered rings, algebras, modules 16W80 Topological and ordered rings and modules
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