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We call a ring R is JN if whose Jacobson radical coincides with upper nilradical, and R is right SR if each element r ∈ R can be written as r = s+r where s is an element from the right socle and r is a regular element of R . SR rings is a class of special subrings of JN rings, which is the extension of soclean rings. We give their some characterizations and examples, and investigate the relationship between JN rings, SR rings and related rings, respectively.

On rings whose Jacobson radical coincides with upper nilradical