We survey some old and recent results concerning the Goldie dimension of modules and modular lattices and its properties which are counterparts of properties of the dimension of linear spaces.

Some properties of the singular ideal are established. In particular its behaviour when passing to one-sided ideals is studied. Obtained results are applied to study some radicals related to the singular ideal. In particular a radical S such that for every ring R, S(R) and R/S(R) are close to being a singular ring and a non-singular ring, respectively, is constructed.

A relation between Goldie dimensions of a modular lattice L and its sublattice LG of fixed points under a finite group G of automorphisms of L is obtained. The method used also gives a relation between ACC (DCC) for L and for LG. The results obtained are applied to rings and modules.

The shape of radicals of semigroups algebras of commutative and cancellative semigroups is studied. The questions asto when a radical of those algebras is homogeneous and if homogeneous radicals have more regular form are examined.

Some characterizations of nil radical and nil semisimple power series rings are given. The upper nil radical of a power series ring in an uncountable set of non-commutative indeterminates is completely described.