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Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring...

Let R be a ring with at least two elements. Prove that M2×2(R)is always a ring (with addition and multiplication of matrices deﬁned as usual).

Solutions

Expert Solution

Let A = M_2x2(R), B = M_2x2(R), C = M₂_x2(R).

i) Now, A+B = + = M₂_x2(R). [Since R is ring]

ii) A+(B+C) = + (+) = + =

And, (A+B)+C = (+)+ = + =

Therefore, A+(B+C) = (A+B)+C for all A,B,C M₂_x2(R).

iii) Now, + = .

Therefore, there exists an element, denoted by O, in M₂_x2(R).such that A+O = A.

Here, O = .

iv) + =

Therefore, for each element A in M₂_x2(R) there exists an element, denoted by -A, in M₂_x2(R) such that A+(-A) = O.

v) A+B = + = = = + = B+A

i.e., A+B = B+A for all A, B M₂_x2(R).

vI) A.B = . = M₂_x2(R).

vii) A.(B.C) = *(*) = =

And, (A.B).C = (*)* = * =

Therefore, A.(B.C) = (A.B).C for A, B, C M₂_x2(R).

viii) A.(B+C) = *(+) = * =

= +

= A.B+A.C

And, (B+C).A = (+)* = * =

= +

= B.A+C.A

Since all conditions are satisfied by M_2x2(R), therefore M_2x2(R) is always a ring.

milcah answered 1 year ago

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