Let M be an R-module
and N ⊂ M. Any H ⊂ M satisfying H + N = M…… (i) and H′⊂ H,
H′ + N = M ⇒ H′ = H…… (ii) will be referred to as a supplement of N in M. In
general N need not have a supplement in M. A module M will be said to have
property (P1) if every N ⊂ M has a supplemenl in M. If for every A ⊂ M, N ⊂ M
with A + N = M, there exists a supplement H of N in M satisfying H ⊂ A, we say
that M has property (P2). Modules with property (P2) play an important role in our
study of dual Goldie dimension. In the present paper we determine the class
of rings R with the property that every M ∈ R-mod possesses property
(P2). These ture out to be left perfect rings. Also the results obtained here
throw more light on the differences between corank and P. Fleury’s spanning
dimension.
