Structure theory of faithful rings. I. Closure operations on lattices

Abstract

A ring having no nonzero left annihilator will be called a faithful ring. Thus, if R is a faithful ring, aRlDO for every nonzero a in R. This series of papers is concerned with the structure theory of faithful rings satisfying various types of minimum conditions. The first paper of the series gives the lattice-theoretic background of our work. It has to do mainly with closure operations on a complete lattice ? having the property that (U i A )) nA = U i (A inA) for every chain { A i } CL and A EL. The lattice of all (right) ideals of a ring has this property. In the first section, the lattices Q(L) and C(L) of all quasi-closure and closure operations respectively on S are defined and discussed. The second section deals with those subsets of L, called insets, that are closed relative to the intersection operation and contain I. If aE C(L), the set La of all A E ? such that Aa =A is an inset of L. The lattice C(L) is shown to be dual isomorphic to the lattice I(L) of all insets of L under the correspondence a--+2a, aE C(L). An m-closure operation on ? is a closure operation that also is an C-endomorphism of L. The set Cm(L) of all m-closure operations on L is a complete sublattice of C(L). If ? is the lattice of all ideals of a ring R and LT is the inset of all semi-prime ideals of R, then wr is an example of an mclosure operation on L. Atomic closure operations on L are studied in the third section. Let CO(2) designate the subset of C(L) containing all a such that GELa. The closure operation aEC0(L) is atomic if La has minimal nonzero elements (atoms). If each nonzero A E La contains an atom of La, a is called homogeneous. The set of all homogeneous m-closure operations on L is shown to be a dual ideal of C? (L). The fourth section is concerned with closure operations on the direct product LX t of the lattices L and M. Let J(L; Ml) (K(L; M)) be the set of all 'U-homomorphisms (n-homomorphisms) of L into M that map 0 into O (I into I). If xEJ(L; M), the inverse mapping x-'EK(1; L), and conversely. The algebraic systems {J(L; M); _, UJ} and {K(MZ; L); _, fl} are isomorphic under the correspondence x--->x-1. Let K'(MZ; L) be the set of all x EK(K(M; L) such that x(U i Mj) = U i xMi for every chain { Mi} c CM. For xEK'(Mr; L), aG C(L), and bEC(T), necessary and sufficient conditions that ax=xb are given. If xEK'(MW; L), the set of all (a, b)EzC(L)XC(QM)