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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.

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.

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