The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This paper focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring [Formula: see text] will be said to be right (respectively, left) nil-ideal-symmetric if [Formula: see text] implies [Formula: see text] (respectively, [Formula: see text]) for nil ideals [Formula: see text] of [Formula: see text]. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property has been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

This article concerns commutative factor rings for ideals contained in the center. A ring R is called CIFC if R/I is commutative for some proper ideal I of R with I ⊆ Z(R), where Z(R) is the center of R. We prove that (i) for a CIFC ring R, W (R) contains all nilpotent elements in R (hence Köthe's conjecture holds for R) and R/W (R) is a commutative reduced ring; (ii) R is strongly bounded if R/N * (R) is commutative and 0 = N * (R) ⊆ Z(R), where W (R) (resp., N * (R)) is the Wedderburn (resp., prime) radical of R. We provide plenty of interesting examples that answer the questions raised in relation to the condition that R/I is commutative and I ⊆ Z(R). In addition, we study the structure of rings whose factor rings modulo nonzero proper ideals are commutative; such rings are called FC. We prove that if a non-prime FC ring is noncommutative then it is subdirectly irreducible.