Question

In: Math

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 = 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.


Related Solutions

. Let φ : R → S be a ring homomorphism of R onto S. Prove...
. Let φ : R → S be a ring homomorphism of R onto S. Prove the following: J ⊂ S is an ideal of S if and only if φ ^−1 (J) is an ideal of R.
Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...
Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.
Let R be a ring and n ∈ N. Let S = Mn(R) be the ring...
Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R. a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R). ii). Show...
In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring...
In this exercise, we will prove the Division Algorithm for polynomials. Let R[x] be the ring of polynomials with real coefficients. For the purposes of this exercise, extend the definition of degree by deg(0) = −1. The statement to be proved is: Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there exist unique polynomials q(x) and r(x) such that f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)). Fix general f (x) and g(x). (a) Let...
Prove that there are domains R containing a pair of elements havig no gcd.(Hint: let R...
Prove that there are domains R containing a pair of elements havig no gcd.(Hint: let R be a subring of F[x], for a field F, where R consistes of all polynomials with no linear terms, then show that x5 and x6 have no gcd) .
For an arbitrary ring R, prove that a) If I is an ideal of R, then...
For an arbitrary ring R, prove that a) If I is an ideal of R, then I[ x] forms an ideal of the polynomial ring R[ x]. b) If R and R' are isomorphic rings, then R[ x] is isomorphic to R' [ x ].
prove that a ring R is a field if and only if (R-{0}, .) is an...
prove that a ring R is a field if and only if (R-{0}, .) is an abelian group
2 Let F be a field and let R = F[x, y] be the ring of...
2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F. (a) Prove that ev(0,0) : F[x, y] → F p(x, y) → p(0, 0) is a surjective ring homomorphism. (b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]} (c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y]...
Prove the following: (a) Let A be a ring and B be a field. Let f...
Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.
Let R be a ring (not necessarily commutative), and let X denote the set of two-sided...
Let R be a ring (not necessarily commutative), and let X denote the set of two-sided ideals of R. (i) Show that X is a poset with respect to to set-theoretic inclusion, ⊂. (ii) Show that with respect to the operations I ∩ J and I + J (candidates for meet and join; remember that I+J consists of the set of sums, {i + j} where i ∈ I and j ∈ J) respectively, X is a lattice. (iii) Give...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT