Isomorphism of direct product of semigroups

Question

I would appreciate some help with the following problem. Consider four semigroups $A,B,C,D$. I was able to prove that $A\cong C\wedge B\cong D$ implies $A\times B\cong C\times D$.

But does also $A\times B\cong C\times D$ imply, that $A\cong C$ and $B\cong D$?

Answer 1

No,it does not.

Let $A=Z_6$,$B=Z_5$,$C=Z_2$ and $D=Z_{15}$ then clearly $A\times B\cong C\times D$ but they are not isomorphic in pairwise.