How much information does the multiplicative semigroup of an algebra contain? How much do we know about an given algebra when we only know its semigroup strucure under the product law?
How far can two algebras be distinguished by knowing only their semigroup strucure?
The same question for rings, special categories of algebras, like *-algebras, Banach algebras, commutative algebras, etc.
Does anybody know results in this direction?
 A: Gluskin [L. M. Gluskin, “Semigroups and rings of endomorphisms of linear spaces”, Izv. Akad. Nauk SSSR Ser. Mat., 23:6 (1959), 841–870 (Russian)] proved that rings of endomorphisms of vector spaces are defined by their multiplicative semigroups. In this paper also some useful information about other rings can be found. It seems there is a thranslation of this article into English.
Addendum: R. E. Peinado, On semigroups admitting ring structure.
Semigroup Forum, 1970, Vol.1, No.1, pp 189-208.
One more addendum: There are some papers in which one proved that topological spaces
are often determined by some subsemigroups of their semigroups of morphisms
[See: Gluskin, L.M.; Schein, B.M.; Shneperman, L.B.; Yaroker, I.S.
Addendum to ”A survey of semigroups of continuous selfmaps”. 
Semigroup Forum, v.14, pp.95-125 (1977)]. Would the spaces have rings of transformations, then those rings will be determined by semigroups. Of course, this is not a case. But maybe these ideas will be usefull for topological algebras.
A: What also came into my mind is, that from matrix algebras of the form $A \otimes M_2$ we can recover the addition from the multiplication:
$$\left (\begin{array} 11 & 1 \\ 0 & 0 \end{array} \right ) \left (\begin{array} aa & 0 \\ b & 0 \end{array} \right ) = \left (\begin{array} aa+b & 0 \\ 0 & 0 \end{array} \right )$$
