## On the Extension to Rings of the Regular Matric Representations |

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annihilator of Q associated factor sets associativity condition automorphism basis characteristic function coefficient field coefficients in f Column commute Corollary correspondence q R(q cr(z»es crossed product crossed-product algebra defined definition of right degree n-1 diagonal diagonal matrix divisor of zero element f element of Q equation ficients field f finite formed by substitution Furthermore Galois group hilator hypothesis inner brackets intermediate right linear invariant factors left anni left multiplications linear algebra linear associative algebra matrices with coefficients minimum function n-rowed square matrices necessary and sufficient non-zero values phism proves the theorem quaternion R-representation Ra(j repre representation of Q right divisor right linear ring right regular representation ring Q rowed square satisfied scalar multiplication set of elements similarity transformation subfield subgroup subring sufficient condition sufficient to guarantee suppose Theorem Theorem 9 thesis tion tj(si(q transposition set zero element