Let M2(Q) be the ring of all 2 × 2 matrices over Q, let R = E11M2(Q), where Eij (1 ≤ i, j ≤ 2) is the martix with 1 in the i−j entry and 0 elsewhere, and let I = QE12.
(3.1) Prove that R is a subring of M2(Q).
(3.2) Prove that I is an ideal in R.
(3.3) Show that IR = {0} and RI ̸= {0}. Deduce from these facts that R is not commutative and has no idenity.
(3.4) Describe the quotient ring R/I and show that R/I is commutative and has a unity.
(3.1) Prove that R is a subring of M2(Q).
(3.2) Prove that I is an ideal in R.
(3.3) Show that IR = {0} and RI ̸= {0}. Deduce from these facts that R is not commutative and has no idenity.
(3.4) Describe the quotient ring R/I and show that R/I is commutative and has a unity.