On varieties of groups generated by wreath products of abelian groups

Generalizing results of Higman and Houghton on varieties generated by wreath products of finite cycles, we prove that the (direct or cartesian) wreath product of arbitrary abelian groups $A$ and $B$ generates the product variety $var(A)\cdot var(B)$ if and only if one of the groups $A$ and $B$ is not of finite exponent, or if $A$ and $B$ are of finite exponents $m$ and $n$ respectively and for all primes $p$ dividing both $m$ and $n$, the factors $B[p^k]\mathrm{/}B[p^{k-1}]$ are infinite, where $B[s]=⟨b\in B\mathrm{\mid }{b}^s=1⟩$ and where $p^k$ is the highest power of $p$ dividing $n$.