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Does the ring of full rank square matrices form a domain? - Answers

It would form a domain, except that it fails to even be a ring. The 0 matrix has rank 0, so it is never a full rank matrix - therefore the set of full rank square matrices doesn't have an additive identity. It is true that there are no zero divisors among the full rank square matrices: if AB=0, and A has full rank, then it's invertible, so A-1AB=A-10, or B=0. Similarly, if BA=0, BAA-1=0A-1 so B=0.



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Does the ring of full rank square matrices form a domain? - Answers

https://math.answers.com/other-math/Does_the_ring_of_full_rank_square_matrices_form_a_domain

It would form a domain, except that it fails to even be a ring. The 0 matrix has rank 0, so it is never a full rank matrix - therefore the set of full rank square matrices doesn't have an additive identity. It is true that there are no zero divisors among the full rank square matrices: if AB=0, and A has full rank, then it's invertible, so A-1AB=A-10, or B=0. Similarly, if BA=0, BAA-1=0A-1 so B=0.



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https://math.answers.com/other-math/Does_the_ring_of_full_rank_square_matrices_form_a_domain

Does the ring of full rank square matrices form a domain? - Answers

It would form a domain, except that it fails to even be a ring. The 0 matrix has rank 0, so it is never a full rank matrix - therefore the set of full rank square matrices doesn't have an additive identity. It is true that there are no zero divisors among the full rank square matrices: if AB=0, and A has full rank, then it's invertible, so A-1AB=A-10, or B=0. Similarly, if BA=0, BAA-1=0A-1 so B=0.

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      It would form a domain, except that it fails to even be a ring. The 0 matrix has rank 0, so it is never a full rank matrix - therefore the set of full rank square matrices doesn't have an additive identity. It is true that there are no zero divisors among the full rank square matrices: if AB=0, and A has full rank, then it's invertible, so A-1AB=A-10, or B=0. Similarly, if BA=0, BAA-1=0A-1 so B=0.

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