\(m\)-quasinormal \(f\)-rings (Q5936007)

Let \(R\) be a lattice-ordered ring. \(R\) is called an \(f\)-ring if \(ca\wedge = ac\wedge b = 0\) whenever \( a\wedge b = 0\) and \(c>0.\) Let \(X\) be a completely regular topological space. Then the set of real-valued continuous functions on \(X\), denoted by \(C(X)\), is an \(f\)-ring. The main result of this paper is Theorem 4.3, which states that \(C(X)\) has the property that the sum of any \(m\) minimal prime ideals is a maximal ideal or the entire ring (called \(F_m\)-space in this paper) if and only if the subspace \(\bigcap_{j=1}^{m} \text{cl(coz}(f_j))\subset X\) is a \(P\)-space for every pairwise disjoint family \(\{f_j\}_{j=1}^{m}\subset C(X).\) This theorem is a generalization of \textit{S. Larson}'s results [Commun. Algebra 25, 3859-3888 (1997; Zbl 0952.06026)]. In section two, the author describes three measures: rank, prime character, and filet character, which are defined on root systems and whose values will determine some portion of the structure of a lattice-ordered group. Based on this section, the author discusses, in section three, \(m\)-quasinormal \(f\)-rings, which are a generalization of quasinormal \(f\)-rings [see, e.g., \textit{S. Larson}, Can. J. Math. 38, 48-64 (1986; Zbl 0588.06011)]. Since \(C(X)\) is a commutative \(m\)-quasinormal \(f\)-ring with identity (\(F_m\)-space), this \(F_m\)-space can be characterized in section four (main results). Furthermore, in section five, the author gives some equivalent conditions under which \(\beta X\) is an \(F_m\)-space for some integer \(m\).