Are There More Zero Divisors in a Ring of 2x2 Matrices Than Initially Thought?

[kuahji]

Let R be a the ring of all 2 X 2 matrices with entries from Z, where the operations are standard matrix addition and standard matrix multiplication, but the entries of the sum and product are determined by addition and multiplication mod 2.

Find all zero divisors.

The professor said there are 9 zero divisors. Below I list the ones I came up with.

The easy ones are (1,0),(0,0) ; (0,1),(0,0) ; (0,0),(1,0) ; (0,0),(0,1) ; (1,0),(1,0) ; (0,1),(0,1)

However my professor says there are three more. My question then becomes technical. Does the zero element (0,0),(0,0) count as a zero divisor? He stated that (0,0),(1,1) is another zero divisor. My problem with this is that it's idempotent. You could do (0,0),(1,1) * (1,1),(1,1) & get the zero matrix in Z mod 2. But by definition of a zero divisor ab=0R, does a & b have to be different? If we don't then we could do (1,1),(1,1) * (1,1),(1,1) would be the zero matrix. So we then just multiplied to zero divisors together & defeated the purpose. Again, it's a technical question.

 

[HallsofIvy]

No, the 0 matrix is not a "zero-divisor". No, a and b do not have to be different. I'm not sure what you mean by "defeated the purpose". What "purpose"? If you are referring to the "(0,0),(1,1)*(1,1)(1,1)" example, you didn't "multiply two zero matrices", in the you multiplied two zero-divisors- and that always happens. a is a "zero-divisors" if and only if it is a non-zero element such that there exist another non-zero element, b, such that ab= 0. And, of course, that implies that b is also a zero-divisor.

 

[kuahji]

Take the unit (0,1),(1,1) for example. If I do (0,1),(1,1) * (1,0),(0,0) = (0,0),(0,0). But by what you just said, then the unit would be a zero divisor, & I though that wasn't possible.

The definition given in the book of a zero divisor is "A nonzero element in a in commutative ring R is a zero divisor if there exists a nonzero element b of R such that ab=0R."

So I guess maybe I just read the definition incorrectly. I didn't realize multiplying two zero divisors together to get the 0 matrix was allowed to show that one or the other was a zero divisor. Namely because of the example I showed above.

*edit* yeah I think that's what it is, I didn't understand it fully & realize now (at least I think), you just not that a zero divisor can not be a unit. But let me finally ask this, if I multiply two matrices together & get the zero matrix, is that enough to show that we have zero divisors (granted we checked that one is not a unit)?

 

[Hurkyl]

No, the 0 matrix is not a "zero-divisor".

Depends on the definition used. With the one he cites, 0 is a zero-divisor. I confess that I didn't realize some people excluded zero until a couple weeks ago!