MATH21112 Rngs and Fields
Example Sheet 2 - Rings and Subrings
1. Let R be the set of functions f : R → R with addition and multiplication defined as
(f + g)(x) = f(x) + g(x) and (fg)(x) = f(x)g(x) for all x ∈ R.
Show that R is a ring by verifying the ring axioms (R1)-(R4).
2. Show that Z[√2] = {a + b√2 | a, b ∈ Z}, with addition and multiplication of real numbers, is a ring. (Hint: show that it is a subring of R).
3. Let SL2 (Z) denote the set of all matrices of the form with
a,b,c, d ∈ Z and ad - bc = 1.
Is this set a ring under matrix addition and multiplication? Is this set a group under matrix multiplication?
4. Consider the ring of matrices
How many elements are there in this ring? Write down some subrings of M2 (Z2 ).
5. Show that the set Q[√2, √3] = {a + b√2 + c√3 + d√6 | a,b,c, d ∈ Q} is a subring of R.
Explain why Q[√2, √3] is the smallest subring of R containing Q, √2 and √3.
(By ‘smallest subring’ here we mean that any other subring S of R which contains Q, √2 and √3 must contain Q[√2, √3].)