Rings in which annihilators form a sublattice of the lattice of ideals
An element of a ring R is said to be regular, if \(a=axa\) for some \(x\in R\). Regular elements of R constitute a semigroup if and only if R is an AC-ring, i.e. left ideals of R generated by single idempotents constitute a sublattice of the lattice of left ideals of R. The equivalence of the following properties of a ring R is stated: (1) R is a left Rickart AC- ring; (2) R is a left Rickart ring and the sum of every two left ideals generated by single idempotents is a left annihilator; (3) R/M(R), where M(R) is the regular radical of R [\textit{B. Brown}, \textit{N. H. McCoy}, Proc. Am. Math. Soc. 1, 165-171 (1950; Zbl 0036.297)], is a left Rickart ring without nilpotent elements and the left ideal Ra is projective for each \(a\in R\) such that \(a+M(R)\) is a non zero-divisor of R/M(R). A ring R is a left Rickart ring with \(r(I\cap J)=r(I)+r(J)\) for every left ideals of R (r(X) means the right annihilator of X) if and only if \(R=Q\oplus B\), where Q is a left self-injective ring and B is a Baer ring with regular maximal ring of quotients. A ring R is a left Rickart ring with \(r(\cap I_ i)=\sum r(I_ i)\) for each set of left ideals of R if and only if \(R=S\oplus T\), where S is a direct sum of integral domains and T is an Artinian semisimple ring.
scientific article; zbMATH DE number 3855312
scientific article