  We define a generic multiplication in quantised Schur algebras and thus
obtain a new algebra structure in the Schur algebras. We prove that via a
modified version of the map from quantum groups to quantised Schur algebras,
defined by A. A. Beilinson, G. Lusztig and R. MacPherson, a subalgebra of this
new algebra is a quotient of the monoid algebra in Hall algebras studied by M.
Reineke. We also prove that the subalgebra of the new algebra gives a geometric
realisation of a positive part of 0-Schur algebras. Consequently, we obtain a
multiplicative basis for the positive part of 0-Schur algebras.
