# semigroup ideals

For the semigroup $$‎S_{3 \times 3} = \bigl\{(a_{ij}) \bigm| a_{ij} \in \mathbb Z_2 = \{0,1\}\bigr\}$$ ‎‎(the set of all $‎3‎\times‎‎3$ matrices with entries from $\mathbb Z_2$) under multiplication. Find all the ideals of $S_{‎3‎\times‎‎3}$.

• ...please...? Any ideas, self work...? – DonAntonio Oct 15 '13 at 10:45

Hint: For any $A \in S_{3\times 3}$ there are $B,C \in {\rm GL}(3, \mathbb Z_2)$ (the invertible elements of $S_{3\times 3}$) such that $$BAC = \begin{pmatrix} 0\mid1 & 0 & 0 \\ 0 & 0\mid1 & 0 \\ 0 & 0 & 0 \mid 1 \end{pmatrix}$$ where the number of ones is determined by the rank of $A$.