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