  Nonassociative algebras satisfying the polynomial identities x(yz)=y(xz) and
(xy)z=(xz)y are called bicommutative. We prove the following results: (i)
Finitely generated bicommutative algebras are weakly noetherian, i.e., satisfy
the ascending chain condition for two-sided ideals. (ii) We give the positive
solution to the Specht problem (or the finite basis problem) for varieties of
bicommutative algebras over an arbitrary field of any characteristic.
