Structure theory of faithful rings. II. Restricted rings

Abstract

The first paper of this series(2) concerned itself mainly with closure operations on a lattice. This paper applies these results to the global structure theory of a faithful ring and its modules. A ring R is called faithful if aR = 0 implies a= 0. Let ? (?2) be the lattice of right (two-sided) ideals of R, and c={A; AC?', AnA =0, A=All}. The set i: becomes a Boolean algebra with the obvious definition of the union operation. In case 3Y is complete, R is called a restricted ring. Being complete, 9Y induces a closure operation f on ? and ?V. It is shown that if f is homogeneous, then there exist irreducible rings Ai such that i AiCR C As, where j ( *) designates the discrete (full) direct sum and 47 is a universal extension ring of Ai. A reducible ring is shown to be restricted, where R is reducible if and only if for every pair A, B of ideals of R with zero intersection, there exist ideals A'DA and B'DB also with zero intersection such that A'nB'I-0. Modules of a faithful ring are studied in the fourth section. It is shown that every suitably restricted closure operation on ? induces a closure operation on Y, the lattice of submodules of a R-module M. If 'A, A CR, designates the annihilator of A in M, then it is shown in the fifth section that 3c 'A; A EDY } is a Boolean algebra. In the final two sections, it is assumed that the ring R and the R-module M have the property that for every nonzero element x in R or M there exists a nonzero A G2 such that xa0 for every nonzero aEA. It is shown that R is restricted, aind that every A in ? (M1Z) has a unique maximal essential extension A, (At). For the closure operation s on ? so defined, 5 is proved to be the center of the lattice 28, and similarly for xC in St. Imbedding ll in its unique minimal injective extension M, it is proved that the lattice o't is isomorphic to the lattice of principal right ideals of the centralizer C of R over M. If s is atomic, e is a full direct sum of primitive rings with minimal right ideals. 1. Faithful rings. If R is a ring, then ?(R)(2'(R)) will designate the lattice of all right (two-sided) ideals of R. We shall upon occasion write S or ?' if the ring in question is obvious. For each subset A of R, AI (Ar) will