Years ago, I read a paper about to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequence of two simpler matrices $S$ and $T$. In my memory, it seems that the length of the product sequence is 3 or 5. That is,

$$X=S^a\cdot T^b\cdot S^c\quad\quad\mbox{or}\quad\quad X=S^a\cdot T^b\cdot S^c\cdot T^d\cdot S^e$$ for some integers $a,b,c,d,e\in\mathbb Z$.

But now, I forget the concrete reference. So, I have no threads to find the method. Is it possible? What are the concrete references?

In addition, is this question a question about representation theory? In other words, is there a representation theory about the matrix ring $M_n(\mathbb Z_N)$?

Years ago, I read a paper about how to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequences of two simpler matrices $S$ and $T$. As I recall, the length of the product sequence is $3$ or $5$. That is,

$$X=S^a\cdot T^b\cdot S^c\quad\quad\mbox{or}\quad\quad X=S^a\cdot T^b\cdot S^c\cdot T^d\cdot S^e$$ for some integers $a,b,c,d,e\in\mathbb Z$.

But now I forget the concrete reference so I have no threads to find the method. Is this representation always possible? What are some references?

In addition, is this a question about representation theory? In other words, is there a representation theory about the matrix ring $M_n(\mathbb Z_N)$?

Years ago, I read a paper about to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequence of two simpler matrices $S$ and $T$. In my memory, it seems that the length of the product sequence is 3 or 5. That is,

$$X=S^a\cdot T^b\cdot S^c\quad\quad\mbox{or}\quad\quad X=S^a\cdot T^b\cdot S^c\cdot T^d\cdot S^e$$

But now, I forget the concrete reference. So, I have no threads to find the method. Is it possible? What are the concrete references?

In addition, is this question a question about representation theory? In other words, is there a representation theory about the matrix ring $M_n(\mathbb Z_N)$?

Years ago, I read a paper about to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequence of two simpler matrices $S$ and $T$. In my memory, it seems that the length of the product sequence is 3 or 5. That is,

$$X=S^a\cdot T^b\cdot S^c\quad\quad\mbox{or}\quad\quad X=S^a\cdot T^b\cdot S^c\cdot T^d\cdot S^e$$ for some integers $a,b,c,d,e\in\mathbb Z$.

But now, I forget the concrete reference. So, I have no threads to find the method. Is it possible? What are the concrete references?

In addition, is this question a question about representation theory? In other words, is there a representation theory about the matrix ring $M_n(\mathbb Z_N)$?

Years ago, I read a paper about to re-write any $d\times d$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequence of two simpler matrices $S$ and $T$. But now, I forget the concrete reference. So, I have no threads to find the method. Is it possible? What are the concrete references?

In addition, is this question a question about representation theory? In other words, is there a representation theory about the matrix ring $M_d(\mathbb Z_N)$?

Years ago, I read a paper about to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequence of two simpler matrices $S$ and $T$. In my memory, it seems that the length of the product sequence is 3 or 5. That is,

$$X=S^a\cdot T^b\cdot S^c\quad\quad\mbox{or}\quad\quad X=S^a\cdot T^b\cdot S^c\cdot T^d\cdot S^e$$

But now, I forget the concrete reference. So, I have no threads to find the method. Is it possible? What are the concrete references?

In addition, is this question a question about representation theory? In other words, is there a representation theory about the matrix ring $M_n(\mathbb Z_N)$?