# a matrice that becomes zero when multiplied with its factor

suppose there is a group of matrices. these matrices multiplied result in a matrix in the group. the matrix can be divided in only one unique way of multiplication of some matrices in the group. we call these factors. when a matrix is multiplied with one of its factor matrices, the mattix becomes zero matrix.

so, is there any group of matrix satisfies this property?

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Restating your question for rings, is there a factorial matrix ring with zero divisors? No, all factorial rings are integral. –  Alexei Averchenko Oct 25 '12 at 15:00

No group of matrices can contain the zero matrix, as the zero matrix does not have an inverse...

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Well the singleton of a zero matrix actually forms a trivial group... –  Marc van Leeuwen Oct 25 '12 at 14:57
Okay, right. I should have written "no nontrivial group of..." –  Sh4pe Oct 25 '12 at 15:05

Your question is rather unclear (the word group is not in place for instance) but you seem to be asking for a subring of a matrix ring that has unique factorization and also zero divisors. However factorzation does not mesh well with the presence of zero divisors: if one factor in a product is a zero divisor, you can always add its complementary zero divisor to the other factor without changing the product, thus ruling out unique factorization. In fact for divisibility questions one almost always supposes the ring is a domain.

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