Left Divisors of Zero in the Ring of Upper Triangular Matrices
Understanding the left divisors of zero in the ring of upper triangular matrices is a fascinating exploration in ring theory. Given the notation begin{bmatrix} Z Z_2 0 Z_2 end{bmatrix}, we are dealing with matrices of the form:
[ M begin{pmatrix} m a 0 b end{pmatrix} ]
where m in Z and ab in Z/2Z end{pmatrix}, and the right column is considered modulo 2 to stay in Z/2Z.
Definition of Left Zero Divisors
A matrix M is a left zero divisor if there exists a non-zero matrix N such that MN 0. Let's explore this concept with a general element N of the form:
[ N begin{pmatrix} n c 0 d end{pmatrix} ]
Product of Matrices
Computing the product MN yields:
[ MN begin{pmatrix} mn overline{mcad} 0 overline{bd} end{pmatrix} ]
where the bars on the right column indicate elements are interpreted modulo 2.
Conditions for Zero Product
The requirement for MN 0 is:
[ begin{pmatrix} 0 bar{0} 0 bar{0} end{pmatrix} ]
This implies the need for certain elements to be zero:
Case 1: m 0
In this case, we have:
[ MN begin{pmatrix} 0 overline{ad} 0 overline{bd} end{pmatrix} ]
Here, we can achieve MN 0 by setting d 0 regardless of a and b. This makes the matrices of the form:
[ M begin{pmatrix} 0 a 0 b end{pmatrix} ]
left zero divisors, including when a b 0.
Case 2: m eq 0 and m text{ is even}
For this case, we need n 0. The product simplifies to:
[ MN begin{pmatrix} 0 overline{0ad} 0 overline{bd} end{pmatrix} ]
By setting d 0, we get zero in both the top-right and bottom-right entries. We can also set c 1. Therefore, we have:
[ begin{pmatrix} 2k a 0 b end{pmatrix}begin{pmatrix} 0 1 0 0end{pmatrix} begin{pmatrix} 0 bar{0} 0 bar{0} end{pmatrix} ]
which shows that matrices of the form:
[ M begin{pmatrix} 2k a 0 b end{pmatrix} ]
are left zero divisors, subsuming the previous case when 2k is arbitrary.
Case 3: m text{ is odd}
Here, we need n 0. The product is:
[ MN begin{pmatrix} 0 overline{cad} 0 overline{bd} end{pmatrix} ]
We need bd 0, meaning either b 0 or d 0 or both. Let's explore the subcases:
Subcase 3.0: b 0
In this subcase, MN begin{pmatrix} 0 overline{cad} 0 0 end{pmatrix} ]. We need overline{cad} 0. Choosing a 0 and c 1, d 1 works, and choosing a 1 and c d 1 also works. Therefore, we have:
[ M begin{pmatrix} 2k 1 a 0 0 end{pmatrix} text{ where } a in Z/2Z text{ and } k in mathbb{Z} ]
Subcase 3.1: b 1
In this subcase, c 1 and MN begin{pmatrix} 0 overline{1ad} 0 0 end{pmatrix} ]. Since d 0, it is impossible for overline{1ad} 0. Therefore, this case does not generate any new left zero divisors.
In conclusion, we have two main types of left zero divisors:
[ M begin{pmatrix} 2k a 0 b end{pmatrix} text{ and } M begin{pmatrix} 2k 1 a 0 0 end{pmatrix} ]
where (a, b in Z/2Z), and (k in mathbb{Z}). This completes our exploration of left zero divisors in the ring of upper triangular matrices.