# Linear Associative Algebras by Alexander Abian (Auth.)

By Alexander Abian (Auth.)

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**Example text**

Thus, (20) is established. Finally, let (20) hold. Let V e T be such that V = Vx + V2 + • • • + Vn = C/i + t/ 2 + • • • + t/ n with, say, Kx 4= C/j and F, e Tx and [/, e r*. Then Kx-C/i 4= 0 and { ( J ^ - f A ) } C ( ^ H 2 ^ j ) contradicting (20). Thus, (18) is established. Hence, Theorem 4 is proved. THEOREM 5. Let °U be a subspace of a vector space T. , y = __
__

__If a ring 52 has at least one nonzero element r then for m > 1 the ring 8%m has divisors of zero. This can be seen from the following example. ° °)(° 1=(° °) (51) { } 0 r)\0 0/ V0 0 | For m > 1, even if Sftm is a commutative ring then only in very excep tional cases is £ftm commutative. In general &m is a noncommutative ring. 2In particular, if @l has a unity element then as (51), (52), (53) and (54) show for m > 1 the ring £ftm has divisors of zero, is not commutative, has nonzero nilpotent elements and has nonzero idempotent elements which are not necessarily equal to the unity of 3fcm. __

Prove that every total m by m matrix ring &m over ^ is simple and that &m&m + {0}. 9. m the total m by m matrix ring over ^ . Prove that Z Y = lm implies YX = Im for every element X and Y of ,^ m where lm is the unity of fflm. 10. , a square matrix such that atj = — aji) over the field of real numbers. Prove that if m is an odd number then ,4 is singular. Prove also that if A has an inverse then the inverse is a skew symmetric matrix. 11. Let 3? be the field of real numbers and ZFm the total m by m matrix ring over 8F.