In: Math
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).
Let A = M2x2(R), B = M2x2(R), C = M2x2(R).
i) Now, A+B = + = M2x2(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 M2x2(R).
iii) Now, + = .
Therefore, there exists an element, denoted by O, in M2x2(R).such that A+O = A.
Here, O = .
iv) + =
Therefore, for each element A in M2x2(R) there exists an element, denoted by -A, in M2x2(R) such that A+(-A) = O.
v) A+B = + = = = + = B+A
i.e., A+B = B+A for all A, B M2x2(R).
vI) A.B = . = M2x2(R).
vii) A.(B.C) = *(*) = =
And, (A.B).C = (*)* = * =
Therefore, A.(B.C) = (A.B).C for A, B, C M2x2(R).
viii) A.(B+C) = *(+) = * =
= +
= A.B+A.C
And, (B+C).A = (+)* = * =
= +
= B.A+C.A
Since all conditions are satisfied by M2x2(R), therefore M2x2(R) is always a ring.