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 $S_{3\times3}$.
2026-03-28 02:21:04.1774664464
semigroup ideals
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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 $A$.