The concept of right (left) quotient (or residual) of an ideal η by anideal ν of an L-subring μ of a ring R is introduced. The right (left) quotients areshown to be ideals of μ . It is proved that the right quotient [η :r ν ] of an idealη by an ideal ν of an L-subring μ is the largest ideal of μ such that[η :r ν ]ν ⊆ η . Most of the results pertaining to the notion of quotients(or residual) of ...

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...