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.