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\(l\)-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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